Sieves in Number Theory

Size: px
Start display at page:

Download "Sieves in Number Theory"

Transcription

1 Sieves in Number Theory Lecture Notes Taught Course Centre 007 Tim Browning, Roger Heath-Brown Tyeset by Sanro Bettin All errors are the resonsibility of the tyesetter. In articular there are some arguments which, as an exercise for the tyesetter, have been fleshe out or re-interrete, ossibly incor- rectly. Tim s lectures were neater an more concise. Corrections woul be gratefully receive at

2 Contents Introuction Sieve of Eratosthenes 5 3 Large Sieve 0 4 Selberg sieve 9 5 Sieve limitations 3 6 Small gas between rimes 37

3 Chater Introuction Sieves can be use to tackle the following questions: i Are there infinitely many rimes such that + is also rime? ii Are there infinitely many rimes such that = n + for some n N? iii Are there infinitely many rimes such that 4 + is also rime? iv Is every sufficiently large n a sum of two rimes? v Is it true that the interval n, n + contains at least one rime for every n N? These roblems are still oen, but, using Sieves methos, some stes towars their solutions have been one. For examle, in 966 Chen rove a weaker version of iv stating that every sufficiently large n is a sum of a rime an a P where P r enotes the numbers that have at most r rime factors. These roblems are also relate to imortant roblems in other Mathematics branches, such as Artin s rimitive root conjecture, which says that, for all a Z with a 0, ±, there exists infinitely many rimes such that a is a rimitive root moulo. Proosition. If iii is true, then Artin s conjecture is true for a =, i.e. there exists infinitely many rimes such that is a rimitive root moulo.

4 Proof. Let = k +, with k N, an q = 4 + = 8k + 5 be rimes. Recall that for all rime r where a { if r ± mo 8, = r if r ±3 mo 8, is the Legenre symbol. Therefore q = an so there oesn t exist any x such that x mo 8. Furthermore, by Fermat s little theorem, 4 = q mo q an so the orer of moulo q must be,, 4,, or 4. It s easily checke that the orer can t be, or 4, an it can t be either because otherwise = q 4 mo q an so k+ mo q. It remains to show that mo q. If it weren t so, we woul have 4k mo q an so there woul be two ossibilities: k mo q or k mo q. The first is imossible for the same reason as before, the secon is imossible because it woul imly that = = = =. q q q q The funamental goal of sieve theory is to rouce uer an lower boun for sets of the tye SA,, z = #{n A n > z }, where A is a finite subset of N, is a subset of the set of rimes P an z > 0. Examles.. Let A = {n N n x} an = { P 3 mo 4}, then SA,, x =#{n x n, P 3 mo 4} #{n x n = a + b for some corime a, b N}, so through this function we can etect sums of two squares.. Let A = {n N n x} an x < z x. Then SA, P, z = #{n x n > z} where πx = #{ P x}. = πx πz, 3

5 3. Let A = {nn n n N, n N }. Then SA, P, x = #{, N P N < < N N} an this is relate to Golbach conjecture. 4

6 Chater Sieve of Eratosthenes The Möbius function is the function µ : N {0, ±} efine by if n =, µn = 0 if P such that n, r if n = r with,..., r istinct rimes. Lemma. For all n N we have µ = n { if n =, 0 otherwise. Proof. Suose n = e er r with,..., r rimes an e,..., e r N. Then µ = r r µ = r = r = 0. r n r Lemma Abel s artial summation formula. Let λ, λ,... be an increasing sequence of real numbers that goes to an c, c,... a sequence of comlex numbers. Let Cx = λ c n x n an φ : [λ, [ R be of class C. Then λ n X c n φλ n = X λ Cxφ x x + CXφX,. for all X λ. Moreover, if CXΦX 0 as X, then c n φλ n = Cxφ x x,. λ n= rovie that either sie is convergent. 5

7 Proof. One has CXφX c n φλ n = c n φx φλ n = λ n X = λ n X X c n φ x x = λ λ λ n x X λ λ n X n X This roves.. To rove. it s enough to let X go to infinity. Cxφ x x. c n φ x x Let Π = Π, z :=,, z A := {n N n A}, for all N. Alying lemma, we can write SA,, z = µ = µ#a. n A, n,π= = n A n,π Π,z.3 Now, suose that there exist X, R an a comletely multilicative function ω, with ω 0, such that Then we can rove the following ω = 0 P \, #A = ω X + R N..4 Theorem A Sieve of Eratosthenes. Let X, R, ω as above an assume furthermore that. R = Oω. k 0 such that Π,z ω log k log z + O 3. y > 0 such that #A = 0 for > y. 6

8 Then we have where SA,, z = XW z + O W z = x + y log z k+ ex log y, log z log z, z ω. Proof. Assume all the hyothesis in the theorem. For all δ > 0, we have F t, z := ω δ t ω, t, Π Π using Rankin s trick. Since + x e x for all x R, using multilicativity of ω we euce that F t, z t δ + ω t ω δ ex = ex δ log t + δ δ Π Π Π ω. δ Now, writing δ = η an using the inequality e x + xe x for x > 0, we see that η = exη log + η log η + ηz η log, since every rime Π, z is less then z. Therefore F t, z t ex η log t ex ω η Π t ex η log t + ω + ηz η Π Π ω log. Now, alying lemma to c = ω log an φx =, we have log x Π,z ω log z log = by hyothesis. Hence Π,x k log log z + O, ω log x log x x + Π,z F t, z t ex η log t + k log log z + kz η log z. 7 ω log log z

9 Choosing η = log z, we obtain F t, z t ex log t log z..5 log z Moreover, by artial summation lemma with c = ω, φx =, we can conclue x that Π, >y ω F t, z F y, z = t = y t log y log z ex k + log z log y log z k+ ex. log z F y, z y y + t ex y log t log z F t, z t t t.6 Finally, by hyothesis 3 an.3-.4 we have SA,, z = Π, y µ#a = = XW z + O X Π, y Π, >y = XW z + O X + y log z where we use hyothesis an µxω µω + log z k+ ex + O Π, y Π, y ω log y log z We aly the revious theorem to the roblem of twine rimes. Corollary A. Brun s theorem. We have, + P < Proof. It follows from a slight moifie version of Theorem A. Corollary A.. For z x 4 log log x, we have φx, z = #{n x n z} <z x µ R, 8

10 Proof. Exercise. Note that Lemma 3 Merten s formula. We have e γ log z, where γ is Euler s constant. z Proof. See Hary Wright, theorem 49. 9

11 Chater 3 Large Sieve Lemma 4. Let F : [0, ] C be a ifferentiable function with continuous erivative. Then, if we exten F by erioicity to all R with erio, we have for all z N z Proof. We have that Therefore a, a,= a F z a F = F α + F 0 F α α + α a 0 F t t. F α α, a α F α + F t t. 3. Now, let δ = z, so that the intervals I = I a := I a δ, a + δ, for z, a an a, =, are all isjoints an containe in [0, ]. Integrating 3. over I, we obtain a α δ F F α α + F t t α I = I I a F α α + I F α α + δ I a I I F t t α F t t, 0

12 since, if α I, then [ a, α] I. Summing over a an an multilying by z we obtain a F z F α α + F t t z a, a,= z a, a,= z F α α + 0 I 0 I F α α. Theorem B Analytic large sieve inequality. Let {a n } n N be a sequence in C, x N an Sα = n x a n e nα, where e β = exπiβ. Then z a, a,= S a z + 4πx a n. Proof. Alying lemma 4 with F α = Sα, we obtain z a, a,= a S z By Parseval s ientity we have that 0 0 Sα α + Sα α = n x a n n x 0 S αsα α. an, since S α = π n x na ne nα, by Cauchy s inequality an Parseval s equality, we get S αsα α a n 4π n a n 4π x a n, 0 n x n x n x that comletes the roof. Remark. Montgomery-Vaughan 974 an Selberg rove ineenently that 4π can be remove from the analytic large sieve inequality. Moreover, is the best ossible coefficient of x.

13 Next we euce a sieve metho from Theorem B. We nee the following lemma about Ramanujan sums. Lemma 5. For all, n N, let c n = a, a,= e na. Then., = c n = c nc n;. c n = D,n µ D D; 3., n = c n = µ. Proof.. By Bézout s ientity we have c n =. By lemma, we have c n = a = D,n a, a, = = s, s,= e e e na = ns na r, r, = r, s, a,= µ = D a, D µd D = µ D D,n e nr + s e nr = c nc n µd D, a D nad e since a e { na 0 if n, = if n. 3. It s a secial case of the revious oint.

14 Theorem C Arithmetic large sieve inequality. Let P an A = {n N n x}. For each, let Ω = {w,,..., w ω, } be a set of ω resiue classes moulo an ut ω = 0 if /. Finally, let SA,, z = {n A n w i, mo i ω Π, z} an SA,, z = #SA,, z. Then SA,, z z + 4πx, Lz where Lz = z µ ω ω. Proof. Let =, t a square-free integer iviing Π, z. By Chinese remainer theorem, for every i = i,..., i t with i j ω j there exists a unique W i, such that 0 W i, < an W i, w ij, j mo j for j t. Let s call ω = t j= ω j the total numbers of the ossible W i, as we vary i. Now let n SA,, z. Then n W i,, = for all an i. Hence, by lemma 5 item 3, we have µ = c n W i, = a, a,= Summing over i an n SA,, z, we euce that µsa,, zω = c n W i, = a, a,= an Wi, e e i awi,. an e n an therefore, by Cauchy-Schwartz inequality, µsa,, zω a, a,= e i awi, a, a,= an e n. 3

15 The first term on the right han sie is awi, e = e a, i a, W i,,w i, a,= a,= = µ W i,,w i, D,W i, W i, = Dµ D D = ω Wi, W ωω D ω i, a = W i,,w i, D = Dµ D D D = ω E = ω ω, c Wi, W i, µeωe E W i, W i, D W i, W i, where we use lemma 5 item. Hence we have µ SA,, z ω ω a, a,= an e n. an this equality is obviously true also if is not square-free or if it oesn t ivie Π, z. Summing over z an alying Theorem B with a n = if n SA,, z, 0 otherwise, we obtain LzSA,, z z + 4πxSA,, z. Given a rime let s efine q to be the smallest ositive integer such that q is not a square moulo or, i.e. the Legenre symbol is equal to. Note that, q being the Legenre symbol comletely multilicative, q P. Moreover, q = if ±3 mo 8, since { if ± mo 8, = 0 if ±3 mo 8. The best result known is q θ+ε for all ε > 0 unconitionally, where θ = 4 e = 0, 56..., while, assuming the Riemann hyothesis, it is q log. This roblem is linke to Artin s conjecture on rimitive roots. Using Theorem C, we can now rove the following corollary. 4

16 Corollary C.. Let ε > 0 an E ε N = #{rimes N q > N ε }. Then E ε N ε. Proof. Since E ε N E ε N if ε < ε, we can suose ε N. Let A = {,..., N }, = { P = n N ε } an Ω = {v mo = }. Thus ω = #Ω = n for all an h := ω ω = + 3 v if. Theorem C imlies that But an so SA,, N N + 4πN N µ h = + 4πN µ E ε N = N, q>n ε N, 3h N, q>n ε + 4πN h N, h. q>n ε E ε NSA,, N 3 + 4πN. 3. Moreover, we have SA,, N = #{n N N, n m = m N } ε #{n = m k N N ε ε / < j < N ε for j k = ε }. 3.3 Inee if n = m k N with N ε ε < j < N ε for j k = ε, then for all k j m we have = for all j k an =, since N m N / ε ε = mn ε an so m N ε. Thus = =. Using the fact that B log log B, the equation 3.3 gives SA,, N,... k n N ε ε < j <N ε N log ε ε N, m k [ ] N > N k ε N ε ε log N,... k N ε ε < j <N ε ε k,... k j <N ε = N log ε ε ε ε log N 3.4 5

17 since log ε > ε log N for N large enough eening on ε. To comlete the roof it is enough to ut together 3. an 3.4. We now woul like to tackle the following questions: for a, b N how likely is it that the conic C a,b := {ax + by = z, x, y, z 0, 0, 0} P Q has a rational oint? If MH is efine as MH = #{a, b N a, b H, C a,b Q }, what is the ratio MH H as H goes to infinity? We are now going to euce by Theorem C a artial answer to this roblem, but first, we nee to state some efinitions an results. by Let K = R or Q for some rime. The Hilbert symbol for K is the function efine a, b K = for all a, b K. Write { x, y, z K 3 \ {0} s.t. ax + by = z, otherwise a, b K = We ll nee the following roerties: { a, b K = Q a, b K = R. Proosition 3. Let K = Q for some rime or R an let a, a, b K. Then:. a, b = b, a. aa, b K = a, b K a, b K bimultilicativity, { a or b > 0, 3. a, b = a, b < 0, 6

18 β 4. If > an a = α u, b = β v for uv, then a, b = αβ u v α, where the last two factors are Legenre symbols. Proof. See 3 of Serre s A course in arithmetic. It s worthwhile to know the following theorem that roves the C a,b satisfy the Hasse rincile. Theorem Hasse-Minkowski. There exists x, y, z Q 3 \ {0} such that ax + by = z iff a, b = an a, b = for all rimes. Proof. See Serre s A course in arithmetic. Now we are reay to rove the following Corollary C.. We have MH H log H ε. Proof. Let M H, H = #{a, b N a H, µa =, b H, C a,b Q }. Clearly, we have M H, H a H µa M a H, where M a H = #{b H a, b = > }. If we efine = { P > }, A = {b H} an Ω = {v mo v, a, v = }, then M a H SA,, z z > 0. Let s now fix a square-free a H an assume H H. Since a is square-free we can write a = α u for u an α {0, }. Thus, by roosition 3 item 4, we have that if >, 7

19 Ω = { v = theorem C, we therefore obtain where an g = +. L a z = v α} an so ω = if α =, 0 otherwise. Alying M a H z + 4πH, L a z µ z, a Now, let ε > 0 an note that + +ε efine ν :=, we have L a z = z, D a, ε + + ε νa a, a, ε + = iff +ε ε + ε = z, a g ε. If we take z = a an we z, a νa. + ε ν + ε Moreover, we have that z = a H H, thus M H, H µa M a H H νa µa H νa + ε. + ε a H a H Hary an Ramanujan rove that a H β νa M H, H H log H β HH log H ε. a H an so we obtain Finally, note that C uv,bq imlies C u,b Q, so, writing a = uv for u square-free, we get MH v H H M v, H H log H ε. Remark C... The result rove in the revious corollary can be imrove. Hooley an Serre rove that H log H MH H log H. In fact, 8

20 Chater 4 Selberg sieve Eratosthenes sieve investigates the function SA,, z = n N,, n,π=, via the equality <z SA,, z = µ = µ#a. n A Π n, Π, where Π = Π, z = The basic sieve roblem is to fin some arithmetic functions µ ± : N R such that { µ if n, Π =, 4. 0 if n, Π > ; so that n, Π µ + n, Π { if n, Π =, 0 if n, Π >, µ #A = µ SA,, z µ + = µ + #A. Π n A n A Π n, Π Writing #A as #A = ωx n, Π + R with ω comletely multilicative, this gives Sa,, z X Π µ + ω + Π 4. µ + R. 4.3 Selberg sieve arose out of an effort to minimize 4.3 subject to 4.. The key iea is to relace µ + by a quaratic form, otimally chosen. We ll nee the following lemmas 9

21 Lemma 6. Let ζ > 0 an {λ i } i N R. Then l Π, l, l<ζ hols for all Π with < ζ if an only if for all l < ζ, l Π. µly l = ωλ µ y l = δ Π, l δ, δ<ζ ωδλ δ δ 4.4 Proof. If y l = δ Π, l δ, δ<ζ l Π, l, l<ζ ωδλ δ δ µly l = l Π, l, l<ζ = δ Π, δ, δ<ζ for all l < ζ, l Π, we have that µl ωδλ δ δ = µ ωλ. δ Π, l δ, δ<ζ ωδλ δ δ = δ Π, δ, δ<ζ ωδλ δ δ µm = µ m δ δ Π, δ, δ<ζ l, l δ ωδλ δ δ µl µm Vice versa, if 4.4 hel for another {y l } l<ζ with {y l } l<ζ {y l } l<ζ, then there woul exist a maximal l < ζ, l Π such that y l y l, an this is a contraiction since 0 = µly l y l = µ ly l y l 0. l Π, l l, l<ζ m δ 4.5 Lemma 7. Let Π an z, ζ > 0. For all a Π, let G a ζ, z = am Π,z, m<ζ gm, with gm the multilicative arithmetic function efine by gm = ωm m Then, if 0 ω <, we have G ζ, z G ζ, z ω. m ω. 0

22 Proof. We have that G ζ, z = m Π,z, m<ζ = l l gl gm = l gl lm Π,m, l =, m Π, m < ζ m < ζ l m Π, m,=l, m<ζ gm = l gm = l gm = G ζ, z l gl m Π, m < ζ l gl, lm Π,m l =, lm <ζ gm glm since gm 0. To conclue the roof it s enough to observe that gl = + g = + ω ω = ω. l We are now reay to rove the following Theorem D Funamental theorem for Selberg sieve. Let z > 0, y > an ω a comletely multilicative arithmetic function such that 0 ω < Ω an #A = ωx + R. Then SA,, z X G y, z + Π,z, <y 3 ν R, where ν =, G y, z = gl = ωl l l Π,z, l< y l gl, ω.

23 Proof. Let {λ } N R with λ = an efine where [a, b] = the inequality 4., inee µ + =,, =[, ] λ λ, ab is the least common multile of a an b. This choice of a,b µ+ satisfies µ + = n, Π [, ] n,π λ λ =, n,π λ λ = n,π an if n, Π = then n, µ + = µ + = λ =. Thus 4.3 hols, that is Π λ 0 Sa,, z X Π = XM + E, µ + ω + Π µ + R 4.3 say. Now, assume that λ = 0 for y. As a consequence we have that µ + = 0 for y. Thus M = Π µ + ω = [, ] Π λ λ ω[, ] [, ] By conition Ω, we can efine gk = ωk k µk 0, we have gk = k ωk = l k k k µ l ω l =, Π,, < y, ω 0 k ω = k µlωl ωk l l k l ωl = µk µl ωl. l k ω λ ω λ, ω,. 0 an, if ωk 0 an = l k Therefore, by Möbius inversion formula, if ω 0 an µ 0, we have k/l µl ωk/l ω = k. gk

24 Thus M =, Π,, < y, ω 0 = l Π, l< y, ωl 0 gl ω λ Π, l, < y ω λ, ω, = ωλ = yl gl, l Π, l< y, ωl 0, Π,, < y, ω 0 ω λ ω λ k, gk 4.6 say. Alying lemma 6 with = an ζ = y, we get = l Π, l< y µly l = l Π, l< y, ωl 0 µly l = l Π, l< y, ωl 0 So, by Cauchy s inequality, we obtain µl gl l Π, l< y = G y, zm, l Π, l< y, ωl 0 y l µl gl. gl y l gl since Π is square-free an by 4.6. Therefore we have M G y,z an the equality hols if an only if the equality hols in Cauchy s inequality, or, in equivalence, if there exists a constant c such that y l gl = cµl gl l Π, s.t. l < y, ωl 0. So, to obtain the best estimate, we have to choose y l = cµlgl an if that hols, alying again lemma 6 with = an η = y, we get = l Π, l< y µly l = l Π, l< y µl gl = cg y, z. Thus to obtain the otimal estimate we have to fin if there exist some λ such that y l = µlgl G y,z for all l < ζ, l Π. So, alying lemma 6 with ζ = y, we fin that the sought 3

25 λ exist an have to be λ = µ ω l Π, l, l< z = µ G g y, z ω µly l = j Π, j< z µ ωg y, z gj = µ G y, z l Π, l, l< z µl gl ω y G, z, 4.7 using the notation of lemma 7. With this choice of λ we have M = G y,z becomes SA, w, z X G y, z + Π, <y µ + R. an 4.3 Therefore, to conclue it s enough to observe that by 4.7 an lemma 7 we have λ since G ζ, z = Gζ, z an so µ + = λ λ for all square-free. =[, ] =[, ] ν ν = a = 3 ν, a a=0 Theorem D can be use to obtain an uer boun for the function φx, z = #{n x n z}. To rove it we ll nee the following lemmas. Lemma 8. Let H k z = l,k=, l<z µl ϕl, where ϕl is the Euler s φ function. Then H k z ϕk k log z. 4

26 Proof. Firstly we rove the statement for k =. We have that H z = l<z µl ϕl = l= h <z, < < h h i = i= h <z, < < h, α i α α h h = κn<z n, where κn = n is the square-free kernel of n. Thus, On the other han, we have an so H z = l<z = l k l k H z = κn<z µn ϕn = l k µl ϕl n,k=, n <z/l n n<z n<z, l=n,k µl ϕl H kz = k n log z. µn ϕn = l k µn ϕn = l k H k z ϕk k n <z/l, n,z/l= µl ϕl H k z l H k z = log z. µln ϕln k ϕk H kz Lemma 9. For all h N we have S = x µ h ν x + log x h, S = x µ hν + log x h, where ν =. Proof. We have that S x µ x hν = xs. 5

27 Moreover, S = x = µ,..., h x,..., h, = h = µ µ h h = = h, i x µ µ h h µ + log x h. x Remark 9.. Using Perron s formula, one can rove that S x log x h, anyway this imrovement oesn t have any effect on our final result about φx, z, in fact that just forces us to use an asymtotic inequality instea of a simle inequality. Now we are reay to rove the following Corollary D.. We have i φx, z x log z + z + log z 3, ii πx x. log x Proof. If we efine A = {n N n x}, we have that φx, z = SA, P, z. Moreover, we have #A = xω + R, with ω = for all an R <. Alying Theorem D with y = z we have x φx, z Gz, z + 3 ν R, where Gz, z = l Π, l<z ωl l Thus, alying lemma 8 with k =, we fin ϕx, z l l ΠP,z, <z ω = µl ϕl. l<z x log z + <z 3 ν µ 6

28 an so to obtain item i it s enough to aly lemma 9. To euce item ii, we have just to observe that by item i we have an choose z = x log x. πx φx, z + πz x log z + O z log z 3 + z Remark D... In the revious corollary we obtaine a better estimate than the one we coul obtain from corollary A.. This is ue to the fact that the main terms of theorems A an B are basically the same, but the error term of the Selberg sieve is much better than the one of the sieve of Eratosthenes. We can also use Theorem D to estimate πx; k, a = #{rimes x a mo k} for given corime a an k. Corollary D.. Let = {rimes x k} an let { if k, ω = 0 otherwise. Then Proof. Exercise. SA,, z k x ϕk log z + Π,z,,k=, <z 3 ν R. Dirichlet theorem of rimes in a rogression assures that πx; k, a goes to infinity as x if k, a = otherwise it s clearly 0 or. k, a = rimes a mo k have analytic ensity, that is ϕk lim s that coincie with arithmetic ensity a mo k s log s πx; k, a lim x πx 7 = ϕk = ϕk In fact, Dirichlet showe that if

29 but be aware that the two statements aren t equivalent. More recisely, we have πx; k, a x ϕk log x with an error term that s not uniform in k. Siegel an Walfisz rove the following result uniform in k. Theorem Siegel-Walfisz. Let a, k =. For all N > 0 there exists a c = cn > 0 such that for any k logx N we have uniformly in k an where li x := x πx; k, a = ϕk li x + O x ex c log x, u log u is the logarithmic integral function. Moreover, if the generalize Riemann hyothesis hols, we have that, for any k uniformly in k. πx; k, a = ϕk li x + O x logkx, x logx, As a consequence of theorem D, we can rove the following corollary, that gives an estimate for πx; k, a that is worse than the revious ones, but that hols for a bigger range of k. Corollary D.3 Brun-Titchmarsh. Let a, k = an k x. Then 4 x x log log x k πx; k, a ϕk log x + O ϕk k log x, k uniformly in k. Proof. Let A = {n x n a mo k} an = { P k}. Then Moreover #A = x k ω πx; k, a SA,, z + z k +. + R, where ω = { if k, 0 if k 8

30 an R <. Hence, by Corollary D. an Lemma 9, we have SA,, z k x ϕk k log z + Π,z,,k=, <z Taking z = x x 5 k k we comlete the roof. 3 ν µ = x ϕk log z + O z log z 3. Remark D.3.. If we coul relace by δ for some δ > 0 in Corollary D.3, we woul have as a consequence that the Lanau-Siegel zeros on t exist. We now state the following theorem. Theorem E Bombieri-Vinograov Theorem, 965. For all A > 0, there exist c = ca > 0 an B = BA > 0 such that for K = x log x B. max a Z/kZ k K li x πx; k, a ϕk C x log x A Proof. See Davenort, Multilicative number Theory it s rove using the large sieve. Combining Theorems D an E, we can stuy Titchmarsh ivisor roblem, that is to comute the orer of the function Sx = x + a, for a N fixe an where n := n. In 930 Titchmarsh was able to rove that Sx = Ox. The following corollary goes beyon that estimate roviing the asymtotic behaviour of Sx. Corollary E.. For all a N, there exists c > 0 such that x log log x Sx = cx + O. log x 9

31 Proof. For all n N we have that where Thus Sx = x = x, a,= +a, +a n = n, n δn = x δn, { if n is a square, 0 otherwise. δ + a = πx;, a + O x, since x δ + a n x δn + a = O x. x Now, let A > 0 an let B = BA > 0 as in Theorem E. Write πx;, a = πx;, a + x, a,= say. Theorem E imlies that S x = xlog x B, a,= = S x + S x, xlog x B, a,= = li x Moreover we have that xlog x B, a,= <t, a,= li x ϕ + for some c > 0. Hence S x = cx + O by 4.8. S x xlog x B, a,= ϕ + O xlog x B x, a,= x log x A πx;, a + O x xlog x B x, a,= πx;, a πx;, a li x ϕ. ϕ = c log t + O, 4.8 x log log x. Finally, Corollary D.3 imlies log x x ϕ log x x log log x, log x 30

32 Chater 5 Sieve limitations The otimization roblem for the uer boun sieve requires minimising the functional Lµ + := X Π,z µ + ω + Π µ + R, subject to µ + n, Π { if n, Π =, 0 if n, Π >. This is almost a roblem of linear rogramming. To obtain a linear rogramming roblem in stanar form, we nee to write µ + = µ + µ + with µ + i 0 an try to minimize the linear functional Lµ + := X Π,z µ + µ + ω + Π µ + + µ + R, subject to µ + µ + n, Π { if n, Π =, 0 if n, Π > an µ + 0, µ

33 Now, efine where δ i,j is Kronecker s elta an c = k X ω + R Π, k=, x = µ +k Π, k=, b = δ,n,π A n;,k = n A { k if n, 0 otherwise, with n A, Π an k =,. Then, what we are trying to minimize is c T x, uner the conitions Ax b an x 0. The ual roblem is to maximize y T b, subject to y 0 an y T A c T. Note that, if the conitions Ax b, x 0, y T A c T an y 0 hol, we have that c T x y T Ax y T b. 5. Moreover, the strong uality theorem assures that there exist x an y such that the equality hols in 5. an clearly those vectors are solutions for the linear rogramming roblem an its ual. Thus, tackling the ual roblem, we can obtain informations about the best uer boun it s ossible to obtain through sieve methos. Now, in this case the ual roblem is maximizing the function uner the conitions X ω y n 0. Jy = R n A, n y n, n A, n,π= y m X ω + R, Note that, taking y n = for any n, we obtain Jy = SA,, z. Moreover, for any subset à A such that X ω à R, 5. 3

34 taking y n = if n Ã, 0 otherwise, we fin Jy = SÃ,, z. Thus, for any à A satisfying 5., we have Lµ + SÃ,, z an it s easy to show that we can ro the conition à A. We now give an examle where the uer boun given by Selberg sieve is otimal. Let Ωn the number of factor of n counte with multilicity an let λn = Ωn be the Liouville function. Set A ± = {n N n x, λn = }. Now, SA +, P, z = #{n A + n z} = #{n x λn =, n z}. Clearly, if z > x 3 we have that SA +, P, z = πx πz = x x log x + O log + Oz. x We now want to fin an uer boun for SA +, P, z using Selberg sieve. We nee the following lemma. Lemma 0. Let Λx = n x λn. Then there exists c > 0 such that Λx E cx, where E c x = x ex c log x. Proof. Let s consier Mertens function Mx = n x µn. It s well known that µn n= = n s. Moreover, using Perron formula, if x isn t an integer we have that ζs Mx = k+i x s πi k i ζs s s, for any k >. Using Cauchy theorem an the zero free region for ζs, one can rove that Mx = OE c x. Moreover, note that if n/l is square-free, then we have λn = 33

35 µn/l = n µn/. Hence Λn = µn/ = µm E c x. n x m x n x Remark 0.. Note that Riemann hyothesis is true if an only if Mx = Ox +ε for any ε > 0 an if an only if Λx = Ox +ε for any ε > 0. Now, let s go back to our sets A ±. We have that #A ± = #{n x λn =, n} = #{m x λm = λ}. Observe that if λm = λ, then λλm = ant it s 0 otherwise. Thus #A ± = λλm m x = [ x ] λ x Λ = x x E + O c, by lemma 0 an if x. Therefore, we have to take X = x Alying Theorem D to this roblem we obtain the remainer term 3 ν R x µ 3 ν E c <y Π, <y <y µ 9ν <y µ E c x an ω = for all., where we use Cauchy s inequality an we are assuming y < x. Now, by lemma 9, we have <y µ 9ν Moreover, we have x µ E c x <y <y x log y ex log 9 y. ex c log x/y c log x/y. 34

36 Thus Π, <y 3 ν R x log 5 y ex c log x/y an taking y E x, we fin 3 ν R x log 5 x ex Π, <y c 4 log x Therefore, taking y E x an z y, theorem D gives us SA + X x, P, z G y, z + O log, x where G y, z G y, y = l Π, l< y by lemma 8. Thus Taking y = E x, we fin ωl l SA +, P, z SA +, P, z an so, since we alreay knew that SA +, P, z = l x log x. ω = µl l< ϕl log y, y x x log y + O log. x x log x + O x log x 3 x x log x + O log + Oz x for z > x 3, we have that with Selberg sieve we are able to rove an otimal uer boun for x z x log x. Therefore Selberg s coefficients µ+ are otimal solutions to the minimization roblem for Lµ + an, corresonly, A + is otimal for the ual roblem. We now turn to the lower boun sieve roblem. It s clear that also this roblem can be exresse as a linear rogramming roblem with the new conition { µ if n, Π =, 0 otherwise. n, Π 35

37 Obviously, the choice µ = 0 for all satisfies this conition, an the corresoning inequality is SA,, z X Π µ ω Π µ R = 0. Now, for A = A an z > x, we have that SA, P, z =, so the coefficients µ = 0 are essentially otimal for our linear rogramming roblem an thus so is A for the ual roblem. In articular, since A an A + have the same inuts, ω, X an OR, it is not ossible for Sieve machinery to istinguish them. Therefore, for x < z < x, log x we can t rove that SA +, P, z x log x through sieve methos an thus that πx x. log x This roblem is ue to the fact that integers n with Ωn are seen by sieves as same as integers n with Ωn. This henomenon is known as Parity roblem an it s a big limitation for sieve methos. To tackle this kin of roblems is therefore necessary to insert some other machinery that oesn t come from sieve methos. 36

38 Chater 6 Small gas between rimes As a consequence of the rime number theorem stating that πx = #{ x} one can rove that an thus N n= n+ n log n N x, log x lim inf n n+ n log n lim su n n+ n log n. The twin rimes conjecture, saying that there are infinitely many rimes such that + is also rime, leas to think that the much weaker statement lim inf n n+ n log n = 0 is true. Hary an Littlewoo were the first to obtain some results in this irection. In 96, they rove that lim inf n uner the generalize Riemann hyothesis. n+ n log n 3 5, years, one of the last being Mayer s roof 986 of lim inf n Other rogresses have been one over the n+ n log n < 4. Finally, in 005 Golston, Pintz an Yilirim manage to rove that this lim inf is 0 an other results towars the twin rime conjecture. The following are results they were able to obtain. 37

39 Theorem. We have lim inf n n+ n log n = 0. Theorem. We have lim inf n n+ n log n log log n < Moreover, enote with BVθ the following statement max max a,q= y x Λn y q x ϕq x θ n y, log A x n a mo for any A > 0. BVθ Note that Bombieri-Vinograov Theorem Theorem E states that BVθ hols for any θ < an that the Elliott-Halberstam conjecture imlies that BVθ hols for any θ <. The following are conitional results rove by Golston, Pintz an Yilirim. Theorem 3. If BVθ hols for some θ >, then lim inf n n+ n <. Theorem 4. If BVθ hols for all θ <, then lim inf n n+ n 0. Theorem 5. If BVθ hols for all θ <, then lim inf n n+ n 6. We are now going to rove the first 4 theorems. The fifth can be obtaine in a similar way with some refinements. Let H > 0 an H [0, H] Z. H is sai amissible if for any rime there exists n such that n + h h H. For examle, {0, } is amissible, but {0,, 4} isn t, since the conition fails for = 3. Clearly if a set H isn t amissible, there aren t infinitely many n such that n + h is rime for all h H. 38

40 If we set k = #H, to verify that H is amissible, it s enough to check the conition for all rimes k. Thus the set H = { i k < < < k } is amissible for any k an so we can fin amissible sets of any carinality. Now, let N N an let H be an amissible set. Define S 0 := logn + h log 3N N<n N h H, n+h rime Clearly, if we were able to rove that S 0 > 0 for infinitely many N, we woul have that there exist infinitely many n such that h H, n+h rime logn + h > log 3N an so that for infinitely many n there exist at least two h such that n + h is rime. Unfortunately this is not the case, since S 0 as N. Thus, we try to sum just on the n that are more likely to give more than one h such that n + h is rime. To o that, we try with Selberg s iea, multilying the summans by n λ an let this being essentially suorte on almost rimes. Therfore, we consier the sum S := logn + h log 3N N<n N h H, n+h rime. h H n+h λ The otimal values of the λ are still not known in this context an so we try to use the Selberg s sieve ones, that are essentially log + κ ζ/ λ µ, log ζ. where log + x := { log x if x, 0 if x 0, an κ is the imension of the sieve. We take κ k = #H an we write κ = k + l, with l 0. As before, if we are able to rove that S > 0 for infinitely many N we ll obtain 39

41 that there exist at least two h such that n + h is rime for infinitely many n. Now, assume that k, l an H [0, H] are fixe an that N 0 on [0, ξ], we can write S in the form S =, ξ λ λ = D ξ Λ D N<n N,, h H n+h N<n N, D h H n+h h H, n+h rime h H, n+h rime where Λ D := [, ]=D λ λ. Since λ, we have that ξ N. Since the λ are suorte logn + h log 3N logn + h log 3N, Λ D #{, [, ] = D} D. Moreover, λ = 0 unless is squarefree an so the same hols for Λ. Let s fix an h 0 H an let H 0 = H \ {h 0 }. We have Sh 0 := Λ D logn + h = Λ D D ξ D ξ N<n N, n+h 0 rime, D h H n+h = D ξ Λ D N+h 0 < N+h 0, D h H 0 h 0 +h log, N+h 0 < N+h 0, D h H h 0+h log since D, = being > N ξ an D ξ. Now, let a,..., a ν0 D be the classes in the set {x mo D h H 0 x h 0 + h 0 mo D} that are corime to D. If D is rime we have ν 0 D #H 0 k k = D k an ν 0 D is clearly multilicative, so ν 0 D D k hols for all square-free D. Moreover, we have Sh 0 = ν 0 D Λ D D ξ j= N+h 0 < N+h 0, a j mo D log. 40

PRIME NUMBERS AND THE RIEMANN HYPOTHESIS

PRIME NUMBERS AND THE RIEMANN HYPOTHESIS PRIME NUMBERS AND THE RIEMANN HYPOTHESIS CARL ERICKSON This minicourse has two main goals. The first is to carefully define the Riemann zeta function and exlain how it is connected with the rime numbers.

More information

Economics 241B Hypothesis Testing: Large Sample Inference

Economics 241B Hypothesis Testing: Large Sample Inference Economics 241B Hyothesis Testing: Large Samle Inference Statistical inference in large-samle theory is base on test statistics whose istributions are nown uner the truth of the null hyothesis. Derivation

More information

Math 5330 Spring Notes Prime Numbers

Math 5330 Spring Notes Prime Numbers Math 5330 Sring 206 Notes Prime Numbers The study of rime numbers is as old as mathematics itself. This set of notes has a bunch of facts about rimes, or related to rimes. Much of this stuff is old dating

More information

6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks

6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks 6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about one-dimensional random walks. In

More information

A Generalization of Sauer s Lemma to Classes of Large-Margin Functions

A Generalization of Sauer s Lemma to Classes of Large-Margin Functions A Generalization of Sauer s Lemma to Classes of Large-Margin Functions Joel Ratsaby University College Lonon Gower Street, Lonon WC1E 6BT, Unite Kingom J.Ratsaby@cs.ucl.ac.uk, WWW home page: http://www.cs.ucl.ac.uk/staff/j.ratsaby/

More information

Factoring Dickson polynomials over finite fields

Factoring Dickson polynomials over finite fields Factoring Dickson polynomials over finite fiels Manjul Bhargava Department of Mathematics, Princeton University. Princeton NJ 08544 manjul@math.princeton.eu Michael Zieve Department of Mathematics, University

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS Contents 1. Moment generating functions 2. Sum of a ranom number of ranom variables 3. Transforms

More information

Lectures on the Dirichlet Class Number Formula for Imaginary Quadratic Fields. Tom Weston

Lectures on the Dirichlet Class Number Formula for Imaginary Quadratic Fields. Tom Weston Lectures on the Dirichlet Class Number Formula for Imaginary Quadratic Fields Tom Weston Contents Introduction 4 Chater 1. Comlex lattices and infinite sums of Legendre symbols 5 1. Comlex lattices 5

More information

Witt#5e: Generalizing integrality theorems for ghost-witt vectors [not completed, not proofread]

Witt#5e: Generalizing integrality theorems for ghost-witt vectors [not completed, not proofread] Witt vectors. Part 1 Michiel Hazewinkel Sienotes by Darij Grinberg Witt#5e: Generalizing integrality theorems for ghost-witt vectors [not complete, not proofrea In this note, we will generalize most of

More information

As we have seen, there is a close connection between Legendre symbols of the form

As we have seen, there is a close connection between Legendre symbols of the form Gauss Sums As we have seen, there is a close connection between Legendre symbols of the form 3 and cube roots of unity. Secifically, if is a rimitive cube root of unity, then 2 ± i 3 and hence 2 2 3 In

More information

Number Theory Naoki Sato

Number Theory Naoki Sato <sato@artofproblemsolving.com> Number Theory Naoki Sato 0 Preface This set of notes on number theory was originally written in 1995 for students at the IMO level. It covers the basic background material

More information

Lecture 21 and 22: The Prime Number Theorem

Lecture 21 and 22: The Prime Number Theorem Lecture and : The Prime Number Theorem (New lecture, not in Tet) The location of rime numbers is a central question in number theory. Around 88, Legendre offered eerimental evidence that the number π()

More information

Introduction to NP-Completeness Written and copyright c by Jie Wang 1

Introduction to NP-Completeness Written and copyright c by Jie Wang 1 91.502 Foundations of Comuter Science 1 Introduction to Written and coyright c by Jie Wang 1 We use time-bounded (deterministic and nondeterministic) Turing machines to study comutational comlexity of

More information

Complex Conjugation and Polynomial Factorization

Complex Conjugation and Polynomial Factorization Comlex Conjugation and Polynomial Factorization Dave L. Renfro Summer 2004 Central Michigan University I. The Remainder Theorem Let P (x) be a olynomial with comlex coe cients 1 and r be a comlex number.

More information

Prime Numbers. Chapter Primes and Composites

Prime Numbers. Chapter Primes and Composites Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

More information

Lecture 17: Implicit differentiation

Lecture 17: Implicit differentiation Lecture 7: Implicit ifferentiation Nathan Pflueger 8 October 203 Introuction Toay we iscuss a technique calle implicit ifferentiation, which provies a quicker an easier way to compute many erivatives we

More information

Lecture L25-3D Rigid Body Kinematics

Lecture L25-3D Rigid Body Kinematics J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L25-3D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general three-imensional

More information

Unit 3. Elasticity Learning objectives Questions for revision: 3.1. Price elasticity of demand

Unit 3. Elasticity Learning objectives Questions for revision: 3.1. Price elasticity of demand Unit 3. Elasticity Learning objectives To comrehen an aly the concets of elasticity, incluing calculating: rice elasticity of eman; cross-rice elasticity of eman; income elasticity of eman; rice elasticity

More information

Integral Regular Truncated Pyramids with Rectangular Bases

Integral Regular Truncated Pyramids with Rectangular Bases Integral Regular Truncate Pyramis with Rectangular Bases Konstantine Zelator Department of Mathematics 301 Thackeray Hall University of Pittsburgh Pittsburgh, PA 1560, U.S.A. Also: Konstantine Zelator

More information

E (log p)2 + Elogplogq = 2xlogx + 0(x), (1)

E (log p)2 + Elogplogq = 2xlogx + 0(x), (1) 374 MA THEMA TICS: P. ERD6iS PROC. N. A. S. 2K 21/8(kk)-1/6 r) where ( is a rime) and w = 6 if = 3, w =2 if> 3;. a runs through the P -21 quadratic residues of that lie between 0 and, while A runs through

More information

1. R In this and the next section we are going to study the properties of sequences of real numbers.

1. R In this and the next section we are going to study the properties of sequences of real numbers. +a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

More information

2.1 Simple & Compound Propositions

2.1 Simple & Compound Propositions 2.1 Simle & Comound Proositions 1 2.1 Simle & Comound Proositions Proositional Logic can be used to analyse, simlify and establish the equivalence of statements. A knowledge of logic is essential to the

More information

The cyclotomic polynomials

The cyclotomic polynomials The cyclotomic polynomials Notes by G.J.O. Jameson 1. The definition and general results We use the notation e(t) = e 2πit. Note that e(n) = 1 for integers n, e(s + t) = e(s)e(t) for all s, t. e( 1 ) =

More information

A 60,000 DIGIT PRIME NUMBER OF THE FORM x 2 + x Introduction Stark-Heegner Theorem. Let d > 0 be a square-free integer then Q( d) has

A 60,000 DIGIT PRIME NUMBER OF THE FORM x 2 + x Introduction Stark-Heegner Theorem. Let d > 0 be a square-free integer then Q( d) has A 60,000 DIGIT PRIME NUMBER OF THE FORM x + x + 4. Introduction.. Euler s olynomial. Euler observed that f(x) = x + x + 4 takes on rime values for 0 x 39. Even after this oint f(x) takes on a high frequency

More information

Number Theory Naoki Sato

Number Theory Naoki Sato <ensato@hotmail.com> Number Theory Naoki Sato 0 Preface This set of notes on number theory was originally written in 1995 for students at the IMO level. It covers the basic background material that an

More information

M147 Practice Problems for Exam 2

M147 Practice Problems for Exam 2 M47 Practice Problems for Exam Exam will cover sections 4., 4.4, 4.5, 4.6, 4.7, 4.8, 5., an 5.. Calculators will not be allowe on the exam. The first ten problems on the exam will be multiple choice. Work

More information

Double Integrals in Polar Coordinates

Double Integrals in Polar Coordinates Double Integrals in Polar Coorinates Part : The Area Di erential in Polar Coorinates We can also aly the change of variable formula to the olar coorinate transformation x = r cos () ; y = r sin () However,

More information

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

More information

Chapt.12: Orthogonal Functions and Fourier series

Chapt.12: Orthogonal Functions and Fourier series Chat.12: Orthogonal Functions and Fourier series J.-P. Gabardo gabardo@mcmaster.ca Deartment of Mathematics & Statistics McMaster University Hamilton, ON, Canada Lecture: January 10, 2011. 1/3 12.1:Orthogonal

More information

Section 3.1 Worksheet NAME. f(x + h) f(x)

Section 3.1 Worksheet NAME. f(x + h) f(x) MATH 1170 Section 3.1 Worksheet NAME Recall that we have efine the erivative of f to be f (x) = lim h 0 f(x + h) f(x) h Recall also that the erivative of a function, f (x), is the slope f s tangent line

More information

MSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUM-LIKELIHOOD ESTIMATION

MSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUM-LIKELIHOOD ESTIMATION MAXIMUM-LIKELIHOOD ESTIMATION The General Theory of M-L Estimation In orer to erive an M-L estimator, we are boun to make an assumption about the functional form of the istribution which generates the

More information

x if x 0, x if x < 0.

x if x 0, x if x < 0. Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

TRANSCENDENTAL NUMBERS

TRANSCENDENTAL NUMBERS TRANSCENDENTAL NUMBERS JEREMY BOOHER. Introduction The Greeks tried unsuccessfully to square the circle with a comass and straightedge. In the 9th century, Lindemann showed that this is imossible by demonstrating

More information

SOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q. 1. Quadratic Extensions

SOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q. 1. Quadratic Extensions SOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q TREVOR ARNOLD Abstract This aer demonstrates a few characteristics of finite extensions of small degree over the rational numbers Q It comrises attemts

More information

Universiteit-Utrecht. Department. of Mathematics. Optimal a priori error bounds for the. Rayleigh-Ritz method

Universiteit-Utrecht. Department. of Mathematics. Optimal a priori error bounds for the. Rayleigh-Ritz method Universiteit-Utrecht * Deartment of Mathematics Otimal a riori error bounds for the Rayleigh-Ritz method by Gerard L.G. Sleijen, Jaser van den Eshof, and Paul Smit Prerint nr. 1160 Setember, 2000 OPTIMAL

More information

The Dirichlet Unit Theorem

The Dirichlet Unit Theorem Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

More information

Differentiability of Exponential Functions

Differentiability of Exponential Functions Differentiability of Exponential Functions Philip M. Anselone an John W. Lee Philip Anselone (panselone@actionnet.net) receive his Ph.D. from Oregon State in 1957. After a few years at Johns Hopkins an

More information

n-parameter families of curves

n-parameter families of curves 1 n-parameter families of curves For purposes of this iscussion, a curve will mean any equation involving x, y, an no other variables. Some examples of curves are x 2 + (y 3) 2 = 9 circle with raius 3,

More information

Primality - Factorization

Primality - Factorization Primality - Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.

More information

An Approach to Optimizations Links Utilization in MPLS Networks

An Approach to Optimizations Links Utilization in MPLS Networks An Aroach to Otimizations Utilization in MPLS Networks M.K Huerta X. Hesselbach R.Fabregat Deartment of Telematics Engineering. Technical University of Catalonia. Jori Girona -. Camus Nor, Eif C, UPC.

More information

Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series

Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series ACA ARIHMEICA LXXXIV.2 998 Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series by D. A. Goldston San Jose, Calif. and S. M. Gonek Rochester, N.Y. We obtain formulas for computing

More information

Sequences of Functions

Sequences of Functions Sequences of Functions Uniform convergence 9. Assume that f n f uniformly on S and that each f n is bounded on S. Prove that {f n } is uniformly bounded on S. Proof: Since f n f uniformly on S, then given

More information

Bounded gaps between primes

Bounded gaps between primes Boune gaps between primes Yitang Zhang It is prove that Abstract lim inf n p n+1 p n ) < 7 10 7, p n is the n-th prime. Our metho is a refinement of the recent work of Golston, Pintz an Yilirim on the

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

Chapter 2 Limits Functions and Sequences sequence sequence Example

Chapter 2 Limits Functions and Sequences sequence sequence Example Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students

More information

Series. Chapter Convergence of series

Series. Chapter Convergence of series Chapter 4 Series Divergent series are the devil, and it is a shame to base on them any demonstration whatsoever. Niels Henrik Abel, 826 This series is divergent, therefore we may be able to do something

More information

Assignment 9; Due Friday, March 17

Assignment 9; Due Friday, March 17 Assignment 9; Due Friday, March 17 24.4b: A icture of this set is shown below. Note that the set only contains oints on the lines; internal oints are missing. Below are choices for U and V. Notice that

More information

The wave equation is an important tool to study the relation between spectral theory and geometry on manifolds. Let U R n be an open set and let

The wave equation is an important tool to study the relation between spectral theory and geometry on manifolds. Let U R n be an open set and let 1. The wave equation The wave equation is an important tool to stuy the relation between spectral theory an geometry on manifols. Let U R n be an open set an let = n j=1 be the Eucliean Laplace operator.

More information

Math 230.01, Fall 2012: HW 1 Solutions

Math 230.01, Fall 2012: HW 1 Solutions Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The

More information

THE PRIME NUMBER THEOREM

THE PRIME NUMBER THEOREM THE PRIME NUMBER THEOREM NIKOLAOS PATTAKOS. introduction In number theory, this Theorem describes the asymptotic distribution of the prime numbers. The Prime Number Theorem gives a general description

More information

Pythagorean Triples Over Gaussian Integers

Pythagorean Triples Over Gaussian Integers International Journal of Algebra, Vol. 6, 01, no., 55-64 Pythagorean Triples Over Gaussian Integers Cheranoot Somboonkulavui 1 Department of Mathematics, Faculty of Science Chulalongkorn University Bangkok

More information

ANALYTIC NUMBER THEORY LECTURE NOTES BASED ON DAVENPORT S BOOK

ANALYTIC NUMBER THEORY LECTURE NOTES BASED ON DAVENPORT S BOOK ANALYTIC NUMBER THEORY LECTURE NOTES BASED ON DAVENPORT S BOOK ANDREAS STRÖMBERGSSON These lecture notes follow to a large extent Davenport s book [5], but with things reordered and often expanded. The

More information

Homework until Test #2

Homework until Test #2 MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

More information

A New Vulnerable Class of Exponents in RSA

A New Vulnerable Class of Exponents in RSA A ew Vulnerable Class of Exponents in RSA Aberrahmane itaj Laboratoire e Mathmatiues icolas Oresme Universit e Caen, France nitaj@math.unicaen.fr http://www.math.unicaen.fr/~nitaj Abstract Let = p be an

More information

2r 1. Definition (Degree Measure). Let G be a r-graph of order n and average degree d. Let S V (G). The degree measure µ(s) of S is defined by,

2r 1. Definition (Degree Measure). Let G be a r-graph of order n and average degree d. Let S V (G). The degree measure µ(s) of S is defined by, Theorem Simple Containers Theorem) Let G be a simple, r-graph of average egree an of orer n Let 0 < δ < If is large enough, then there exists a collection of sets C PV G)) satisfying: i) for every inepenent

More information

DIRICHLET PRIME NUMBER THEOREM

DIRICHLET PRIME NUMBER THEOREM DIRICHLET PRIME NUMBER THEOREM JING MIAO Abstract. In number theory, the rime number theory describes the asymtotic distribution of rime numbers. We all know that there are infinitely many rimes,but how

More information

CHAPTER 5 : CALCULUS

CHAPTER 5 : CALCULUS Dr Roger Ni (Queen Mary, University of Lonon) - 5. CHAPTER 5 : CALCULUS Differentiation Introuction to Differentiation Calculus is a branch of mathematics which concerns itself with change. Irrespective

More information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

More information

f(x) = a x, h(5) = ( 1) 5 1 = 2 2 1

f(x) = a x, h(5) = ( 1) 5 1 = 2 2 1 Exponential Functions an their Derivatives Exponential functions are functions of the form f(x) = a x, where a is a positive constant referre to as the base. The functions f(x) = x, g(x) = e x, an h(x)

More information

Notes on tangents to parabolas

Notes on tangents to parabolas Notes on tangents to parabolas (These are notes for a talk I gave on 2007 March 30.) The point of this talk is not to publicize new results. The most recent material in it is the concept of Bézier curves,

More information

10.2 Systems of Linear Equations: Matrices

10.2 Systems of Linear Equations: Matrices SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix

More information

On the number-theoretic functions ν(n) and Ω(n)

On the number-theoretic functions ν(n) and Ω(n) ACTA ARITHMETICA LXXVIII.1 (1996) On the number-theoretic functions ν(n) and Ω(n) by Jiahai Kan (Nanjing) 1. Introduction. Let d(n) denote the divisor function, ν(n) the number of distinct prime factors,

More information

Riesz-Fredhölm Theory

Riesz-Fredhölm Theory Riesz-Fredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A Ascoli-Arzelá Result 18 B Normed Spaces

More information

Lecture 13: Differentiation Derivatives of Trigonometric Functions

Lecture 13: Differentiation Derivatives of Trigonometric Functions Lecture 13: Differentiation Derivatives of Trigonometric Functions Derivatives of the Basic Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the

More information

19.2. First Order Differential Equations. Introduction. Prerequisites. Learning Outcomes

19.2. First Order Differential Equations. Introduction. Prerequisites. Learning Outcomes First Orer Differential Equations 19.2 Introuction Separation of variables is a technique commonly use to solve first orer orinary ifferential equations. It is so-calle because we rearrange the equation

More information

The Cubic Equation. Urs Oswald. 11th January 2009

The Cubic Equation. Urs Oswald. 11th January 2009 The Cubic Equation Urs Oswald th January 009 As is well known, equations of degree u to 4 can be solved in radicals. The solutions can be obtained, aart from the usual arithmetic oerations, by the extraction

More information

f(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.

f(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line. Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,

More information

ON DIVISORS OF LUCAS AND LEHMER NUMBERS

ON DIVISORS OF LUCAS AND LEHMER NUMBERS This aer submitte to Acta Mathematica. ON DIVISORS OF LUCAS AND LEHMER NUMBERS C.L. STEWART 1. Introuction Let u n be the n-th term of a Lucas sequence or a Lehmer sequence. In this article we shall establish

More information

MATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.

MATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada. MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division.

More information

Point Location. Preprocess a planar, polygonal subdivision for point location queries. p = (18, 11)

Point Location. Preprocess a planar, polygonal subdivision for point location queries. p = (18, 11) Point Location Prerocess a lanar, olygonal subdivision for oint location ueries. = (18, 11) Inut is a subdivision S of comlexity n, say, number of edges. uild a data structure on S so that for a uery oint

More information

Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS

Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection

More information

FREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES

FREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES FREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES AVNER ASH, LAURA BELTIS, ROBERT GROSS, AND WARREN SINNOTT Abstract. We consider statistical roerties of the sequence of ordered airs obtained by taking

More information

The Impact of Forecasting Methods on Bullwhip Effect in Supply Chain Management

The Impact of Forecasting Methods on Bullwhip Effect in Supply Chain Management The Imact of Forecasting Methos on Bullwhi Effect in Suly Chain Management HX Sun, YT Ren Deartment of Inustrial an Systems Engineering, National University of Singaore, Singaore Schoo of Mechanical an

More information

Inverse Trig Functions

Inverse Trig Functions Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that

More information

MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear: MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

More information

Fourier Analysis of Stochastic Processes

Fourier Analysis of Stochastic Processes Fourier Analysis of Stochastic Processes. Time series Given a discrete time rocess ( n ) nz, with n :! R or n :! C 8n Z, we de ne time series a realization of the rocess, that is to say a series (x n )

More information

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

More information

VERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS

VERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS VERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS MICHAEL DRMOTA, OMER GIMENEZ, AND MARC NOY Abstract. We show that the number of vertices of a given degree k in several kinds of series-parallel labelled

More information

1 Gambler s Ruin Problem

1 Gambler s Ruin Problem Coyright c 2009 by Karl Sigman 1 Gambler s Ruin Problem Let N 2 be an integer and let 1 i N 1. Consider a gambler who starts with an initial fortune of $i and then on each successive gamble either wins

More information

Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes

Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes previous inex next Calculating Viscous Flow: Velocity Profiles in Rivers an Pipes Michael Fowler, UVa 9/8/1 Introuction In this lecture, we ll erive the velocity istribution for two examples of laminar

More information

Here the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and

Here the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and Lecture 9 : Derivatives of Trigonometric Functions (Please review Trigonometry uner Algebra/Precalculus Review on the class webpage.) In this section we will look at the erivatives of the trigonometric

More information

Elementary Number Theory: Primes, Congruences, and Secrets

Elementary Number Theory: Primes, Congruences, and Secrets This is age i Printer: Oaque this Elementary Number Theory: Primes, Congruences, and Secrets William Stein November 16, 2011 To my wife Clarita Lefthand v vi Contents This is age vii Printer: Oaque this

More information

Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year , First Semester Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

More information

A Note on Integer Factorization Using Lattices

A Note on Integer Factorization Using Lattices A Note on Integer Factorization Using Lattices Antonio Vera To cite this version: Antonio Vera A Note on Integer Factorization Using Lattices [Research Reort] 2010, 12 HAL Id: inria-00467590

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

Congruences. Robert Friedman

Congruences. Robert Friedman Congruences Robert Friedman Definition of congruence mod n Congruences are a very handy way to work with the information of divisibility and remainders, and their use permeates number theory. Definition

More information

arxiv:1309.1857v3 [gr-qc] 7 Mar 2014

arxiv:1309.1857v3 [gr-qc] 7 Mar 2014 Generalize holographic equipartition for Friemann-Robertson-Walker universes Wen-Yuan Ai, Hua Chen, Xian-Ru Hu, an Jian-Bo Deng Institute of Theoretical Physics, LanZhou University, Lanzhou 730000, P.

More information

MATH 461: Fourier Series and Boundary Value Problems

MATH 461: Fourier Series and Boundary Value Problems MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter

More information

Large Sample Theory. Consider a sequence of random variables Z 1, Z 2,..., Z n. Convergence in probability: Z n

Large Sample Theory. Consider a sequence of random variables Z 1, Z 2,..., Z n. Convergence in probability: Z n Large Samle Theory In statistics, we are interested in the roerties of articular random variables (or estimators ), which are functions of our data. In ymtotic analysis, we focus on describing the roerties

More information

A Modified Measure of Covert Network Performance

A Modified Measure of Covert Network Performance A Modified Measure of Covert Network Performance LYNNE L DOTY Marist College Deartment of Mathematics Poughkeesie, NY UNITED STATES lynnedoty@maristedu Abstract: In a covert network the need for secrecy

More information

AND. Dimitri P. Bertsekas and Paul Tseng

AND. Dimitri P. Bertsekas and Paul Tseng SET INTERSECTION THEOREMS AND EXISTENCE OF OPTIMAL SOLUTIONS FOR CONVEX AND NONCONVEX OPTIMIZATION Dimitri P. Bertsekas an Paul Tseng Math. Programming Journal, 2007 NESTED SET SEQUENCE INTERSECTIONS Basic

More information

2 HYPERBOLIC FUNCTIONS

2 HYPERBOLIC FUNCTIONS HYPERBOLIC FUNCTIONS Chapter Hyperbolic Functions Objectives After stuying this chapter you shoul unerstan what is meant by a hyperbolic function; be able to fin erivatives an integrals of hyperbolic functions;

More information

Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

More information

61. REARRANGEMENTS 119

61. REARRANGEMENTS 119 61. REARRANGEMENTS 119 61. Rearrangements Here the difference between conditionally and absolutely convergent series is further refined through the concept of rearrangement. Definition 15. (Rearrangement)

More information

The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.

The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors. The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,

More information

Chapter 13: Basic ring theory

Chapter 13: Basic ring theory Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring

More information

Answers to the Practice Problems for Test 2

Answers to the Practice Problems for Test 2 Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan

More information

Lecture Notes: Discrete Mathematics

Lecture Notes: Discrete Mathematics Lecture Notes: Discrete Mathematics GMU Math 125-001 Sring 2007 Alexei V Samsonovich Any theoretical consideration, no matter how fundamental it is, inevitably relies on key rimary notions that are acceted

More information

Algorithms for Constructing Zero-Divisor Graphs of Commutative Rings Joan Krone

Algorithms for Constructing Zero-Divisor Graphs of Commutative Rings Joan Krone Algorithms for Constructing Zero-Divisor Grahs of Commutative Rings Joan Krone Abstract The idea of associating a grah with the zero-divisors of a commutative ring was introduced in [3], where the author

More information