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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

1 ACA ARIHMEICA LXXXIV 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 mean values of Dirichlet polynomials that have more terms than the length of the integration range. hese formulas allow one to compute the contribution of off-diagonal terms provided one knows the correlation functions for the coefficients of the Dirichlet polynomials. A smooth weight is used to control error terms, and this weight can in typical applications be removed from the final result. Similar results are obtained for the tails of Dirichlet series. Four examples of applications to the Riemann zeta-function are included.. Introduction and statement of results. Let {a n } n= be a sequence of real or complex numbers such that for any ε >, a n ε n ε as n. Let s = σ + it be a complex variable and let As = a n n s be a Dirichlet polynomial. By Montgomery and Vaughan s mean value theorem [4] we have As 2 dt = a n 2 n 2σ + On. It immediately follows that if N = o as, then As 2 dt a n 2 n 2σ. On the other hand, if N and σ <, the O-terms in can dominate so that we lose the asymptotic formula. he situation is similar for the 99 Mathematics Subject Classification: Primary M6; Secondary M26. he work of both authors was partially supported by grants from NSF. [55]

2 56 D. A. Goldston and S. M. Gonek mean-square of the tail Dirichlet series A s = a n n s n>n when σ >. Our purpose in this paper is to determine the mean-square behavior of As and A s even when N is significantly larger than. If we square out and integrate termwise in, we see that the O-terms on the right-hand side come from off-diagonal terms. It is these we must carefully estimate therefore when N is large. We treat them by appealing to good uniform estimates for the coefficient correlation functions Ax, h = n x a n a n+h. Such estimates are available for a n, a n = dn the divisor function, a n = µ 2 n the square of the Möbius function, when a n is the nth Fourier coefficient of a modular form, and for a number of other arithmetical functions. Moreover, it is interesting to note that we can often formulate a conjectural estimate for Ax, h even when we cannot estimate Ax, h rigorously. In such cases we can then use our theorems to deduce conditional mean value formulae for the associated Dirichlet series. Since it is no more difficult to treat the more general means 2 and 3 AsBs dt A sb s dt, where Bs = b nn s and B s = n>n b nn s, we shall do so. he precise assumptions we shall make about the sequences {a n } n= and {b n } n= are: A For every ε > we have a n, b n ε n ε. A 2 If Ax = n x a n and Bx = n x b n, then for x we may write 4 Ax = M x + E x and 5 Bx = M 2 x + E 2 x, where 6 7 M x, M 2x ε x + ε, M x, M 2 x ε x + ε

3 and 8 for some θ [,. Mean value theorems 57 E x, E 2 x x + θ A 3 he coefficient correlation functions are of the form C x, h = n x a n b n+h and C 2 x, h = n x b n a n+h 9 C i x, h = M i x, h + E i x, h i =, 2 for x, where M i x, h i =, 2 is twice differentiable for each h =, 2,..., and E i x, h x + ϕ i =, 2 uniformly for h x η for some ϕ [, and some η,. Sometimes we shall also assume A 4 For every ε > we have uniformly for x and h =, 2,... M ix, h ε h ε x + ε i =, 2 Instead of estimating 2 and 3 directly, we find it more advantageous to estimate the integrals I = and 2 I = Ψ U t As Ψ U t A s N N M xx s dx Bs M xx s dx B s N N M 2 xx s dx dt M 2 xx s dx dt. Here M x and M 2 x are as in 4 and 5 and Ψ U t is a real-valued weight function satisfying the following conditions. Let B >, U log B, and C C 2, where C and C 2 are bounded but may be functions of U. hen Ψ U t is supported on [C U, C 2 + U ], 3 Ψ U t = if C + U t C 2 U, and 4 Ψ j U t U j for j =,,...

4 58 D. A. Goldston and S. M. Gonek Note that 3 is vacuous if, for example, C = C 2. he removal of Ψ U from I and I is usually straightforward and will be demonstrated in the examples at the end of the paper. Before stating our results we introduce some more notation and useful estimates. We use ε to represent an arbitrarily small positive number which is fixed during the course of each proof. We then set τ = ε. We always assume that σ, the real part of s, is bounded above and below. he constants implied by the symbols O and may depend on ε, the upper and lower bounds for σ, and other parameters, but never on or parameters dependent on, like N and τ. hus, in particular, our O-terms hold uniformly for bounded σ. We define the Fourier transform of Ψ U t by Ψ U ξ = Ψ U teξt dt, where ex = e 2πix. It follows easily that t 5 Ψ U eξt dt = Ψ U ξ and, since Ψ U t is real, that 6 ΨU ξ = Ψ U ξ. Observe that Ψ U and Ψ U are trivially C 2 C + U. Also, integrating by parts j times and using 4, we see that Ψ U ξ and Ψ U ξ are C 2 C + U U/2πξ j if ξ. hus, for j arbitrarily large we have 7 ΨU ξ, Ψ U ξ C 2 C + U min, U/2πξ j. It follows that 8 ΨU ξ ξ D for ξ ε and that 9 ΨU ξ D 2 for ξ τ, where D and D 2 are arbitrarily large constants. We write K σ x, u = K σ x, u,, U = x 2σ + u σ ΨU x 2π log + u x and easily find by 7 that 2 K σ x, u x 2σ

5 Mean value theorems 59 and 2 x K σx, u x 2σ ε for u/x τ. By the mean value theorem of differential calculus and 7 we have 22 ΨU 2π log + u x = Ψ u U + O +2ε 2πx when u/x τ. Using this and 7 it is not difficult to deduce the approximation u 23 K σ x, u = x 2σ ΨU + Ox 2σ +2ε 2πx for u/x τ. We can now state our main results. heorem. Let σ < σ 2 <, let < ε < /2 be fixed, and let τ = ε. Suppose that the sequences {a n } n= and {b n } n= satisfy A, A 2, and A 3 and that 24 τ N τ / η, where η is as in A 3. Set 25 H = N/τ +. hen 26 I = Ψ U t As = Ψ U + H H h H τ τ N u N u N a n b n n 2σ + N h hτ M xx s dx Bs h H N h hτ M 2 x, hk σx, h dx M x + um 2 xk σx, u dx du M 2 x + um xk σ x, u dx du + ON 2σ+maxθ,ϕ+5ε + O 2ε uniformly for σ σ σ 2 and sufficiently large. N M 2 xx s dx dt M x, hk σ x, h dx

6 6 D. A. Goldston and S. M. Gonek he following weaker version of heorem is easier to apply and sufficient for many applications. Corollary. Let the hypotheses and notation be the same as in heorem except now assume that N and that A 4 also holds. Write 2 2σ C = 2π. 2π hen 27 I = Ψ U a n b n n 2σ + C h M h 2πv, h 2σ Ψ U vv 2σ 2 dv /2πN + C h 2πNv/ /2πN h 2πNv/ 2πNv/ M /2πτN 2C M 2 h h 2πv, h 2σ Ψ U vv 2σ 2 dv u u M 2 u 2σ du 2πv 2πv Re Ψ U vv 2σ 2 dv + O N 2 2σ+5ε + ON 2σ+maxθ,ϕ+5ε + ON 2ε uniformly for σ σ σ 2 and sufficiently large. heorem 2. Suppose that the sequences {a n } n= and {b n } n= satisfy A, A 2, and A 3. Let < σ < σ 2, let < ε < /2 be fixed, and set τ = ε. For σ σ σ 2 write λ = 2σ 2σ 2 ; let 28 τ N τ ε+η/λ η, where η is as in A 3, and set 29 = τ λ N λ/ ε. hen 3 I = Ψ U t A s = Ψ U N<n N M xx s dx B s a n b n n 2σ N M 2 xx s dx dt

7 + + h maxn,hτ h maxn,hτ maxn, maxn, Mean value theorems 6 M x, hk σ x, h dx M 2 x, hk σx, h dx M x + um 2 xk σx, u dx du M 2 x + um xk σ x, u dx du + ON 2σ+maxθ,ϕ+5ε + O ε/2 N 2σ uniformly for σ σ σ 2 and sufficiently large. A simpler form of heorem 2 is provided by Corollary 2. Let the hypotheses and notation be the same as in heorem 2 except now assume that N and that A 4 also holds. hen 3 I = Ψ U a n b n n 2σ N<n /2πN + C h h 2πv, h 2σ Ψ U vv 2σ 2 dv h M /2πN + C + C H + C H 2C H /2πN /2πN /2πN h M 2 h h 2πv, h 2σ Ψ U vv 2σ 2 dv 2πNv/ <h M M 2 /2πN 2πNv/ <h /2πN H M 2πNv/ h h 2πv, h 2σ Ψ U vv 2σ 2 dv h h 2πv, h 2σ Ψ U vv 2σ 2 dv u u M 2 u 2σ du 2πv 2πv Re Ψ U vv 2σ 2 dv + O N 2 2σ+5ε + ON 2σ+maxθ,ϕ+5ε + O ε/2 N 2σ uniformly for σ σ σ 2 and sufficiently large, where C is defined in Corollary.

8 62 D. A. Goldston and S. M. Gonek Although we could make the next theorem more precise by arguing along the lines of the proofs of heorems and 2, the version below is usually all that we require. heorem 3. Assume that the sequences {a n } n= and {b n } n= satisfy A and 6 and that N. Let σ < σ 2 <, < σ < σ 2, s = σ + it, s = σ + it, and let < ε < /2 be arbitrary. hen J = Ψ U t As N 2 σ σ +5ε N M xx s dx B s N M 2 xx s dx dt uniformly for σ σ σ 2 and σ σ σ 2 and sufficiently large. One measure of the strength of our results is how much larger than we may take N and still retain an asymptotic formula. his is determined by the parameters θ, ϕ, and η as can be seen, for example, from 24 and the error term 32 ON 2σ+maxθ,ϕ+5ε in 26 of heorem. It turns out that this term comes from using the pointwise upper bounds for E i x and E i x, h i =, 2 given in A 2 and A 3 to estimate various expressions involving these functions. It is worth noting that if E i x and E i x, h i =, 2 act like random variables in x and behave independently as functions of h, then one might expect to be able to replace 32 by 33 O /2 N /2 2σ+maxθ,ϕ+5ε. his observation makes it easy to conjecture the mean values of very long Dirichlet polynomials as we shall illustrate in Example 3 of Section 5. We would similarly expect 33 to replace the next-to-last error term in heorem 2 and Corollaries and 2. It is also worth noting that one can sometimes exploit averages of E i x, h over h to improve Proof of heorem and its corollary. Multiplying out in, we obtain t I = Ψ U t AsBs dt Ψ U N As M 2 xx s dx dt Ψ U t Bs N M xx s dx dt

9 or Mean value theorems 63 t + Ψ U N N M xx s dx M 2 yy s dy dt, 34 I = I I 2 I 3 + I 4. First consider I. By 5 and 6 we have or I = Ψ U a n b n n 2σ + a n b m mn σ ΨU 2π log m n n<n n<m N + b n a m mn σ Ψ U 2π log m, n n<n n<m N 35 I = Ψ U a n b n n 2σ + I 2 + I 3 for short. In I 2 we set m = n + h and note that by A and 9 the total contribution of those terms with h > n/τ is no more than O, say. It follows that I 2 = n<n h minn/τ,n n + O = n<n h minn/τ,n n a n b n+h n 2σ + h σ ΨU n a n b n+h K σ n, h + O. 2π log Changing the order of summation, we obtain I 2 = a n b n+h K σ n, h + O. h H hτ h By 9 and Stieltjes integration this becomes I 2 = h H + h H N h hτ N h M x, hk σ x, h dx hτ K σ x, h de x, h + O. + h n

10 64 D. A. Goldston and S. M. Gonek he second term equals N h 36 E x, hk σ x, h N h hτ E x, h x K σx, h dx. h H o bound this we use, but first we must check that h x η whenever h H and x [hτ, N h]. his will be the case if h hτ η for every h H, or if hτ H τ η/ η. But this follows from 24 and 25. By, 2, and 2 we now find that 36 is ε hτ ϕ 2σ+ε/2 + N ϕ 2σ+ε/2. h H Here we have appealed to the estimate 37a B A x λ dx A +λ+δ + B +λ+δ, which holds uniformly for A B and bounded λ, where δ > is arbitrarily small, and where the implied constant depends at most on δ. We also note for later use that the δ is unnecessary if λ is bounded away from. Next, using the discrete analogue of this, namely 37b h λ A +λ+δ + B +λ+δ, A h B we see that the sum above is since ε τ ϕ 2σ+ε H +ϕ 2σ+ε + + ε HN ϕ 2σ+ε ε τ N +ϕ 2σ+ε + τ +ϕ 2σ+ε N +ϕ 2σ+5ε + τ +ϕ 2σ+5ε, τ ε < ε < τ 2ε < N 2ε when < ε < /2. If + ϕ 2σ + 5ε, this is and in the opposite case it is N +ϕ 2σ+5ε, because τ N. hus, 36 is N +ϕ 2σ+5ε + and it follows that I 2 = h H N h hτ M x, hk σ x, h dx + O N +ϕ 2σ+5ε + O. reating I 3 in the same way, we obtain I 3 = h H N h hτ M 2x, hk σ x, h dx + O N +ϕ 2σ+5ε + O.

11 Mean value theorems 65 Combining these results with 35, we now find that 38 or I = Ψ U + h H + h H a n b n n 2σ N h hτ N h hτ Next we treat I 2. By 5 we have I 2 = = a n N n σ a n N n σ M x, hk σ x, h dx M 2 x, hk σx, h dx + ON +ϕ 2σ+5ε + O. M 2 xx σ ΨU 2π log x n 39 I 2 = I 2 + I 22. n + n dx M 2 xx σ ΨU 2π log x dx n In I 2 we set x = n + u and note as before that by A, 6, and 9, that portion of the integral with u > n/τ contributes a negligible amount. hus we find that I 2 = minn/τ,n n a n n σ M 2 n + un+u σ ΨU 2π log + u du n + O = a n minn/τ,n n M 2 n + uk σn, u du + O, say. Changing the order of summation and integration, we find that I 2 = τ + H u τ a n M 2 n + uk σn, u du u a n M 2 n + uk σn, u du + O. he first term is τ n ε 2σ du τ N 2σ+2ε + N 2σ+4ε + 2ε

12 66 D. A. Goldston and S. M. Gonek by A, 6, 2 and 37a. By 4 and Stieltjes integration the second term equals 4 H τ N u M xm 2 x + uk σx, u dx du + H τ N u M 2 x + uk σx, u de x du. Integrating by parts and using 6 8, 2, 2, and 37b, we see that the second term is = H M 2 x + uk σx, ue x N u du τ H N u M 2 x + uk σx, u + M 2 x + u x K σx, u E x dx du τ ε H ε τ N τ θ 2σ+ε/2 + N θ 2σ+ε/2 du v θ 2σ+ε/2 dv + ε HN θ 2σ+ε/2 ε τ N +θ 2σ+ε + N +θ 2σ+5ε + 2ε. 4 hus we have I 2 = H τ N u M xm 2 x + uk σx, u dx du + O N +θ 2σ+5ε + O +2ε. We treat I 22 similarly. Setting x = n u, we see that I 22 = n a n n σ M 2 n un u σ ΨU 2π log u du. n Using A, 6, and 9 for that part of the integral for which u > n/τ +, we find that minn/τ+,n a n n σ M 2 n un u σ ΨU 2π log u du n + O = 2 a n n/τ+ M 2 n uk σn u, u du + O.

13 Mean value theorems 67 If we change the order of summation and integration we obtain I 22 = τ + H 2 τ a n M 2 n uk σn u, u du + a n M 2 n uk σn u, u du + O. As in the case of I 2, the first term is easily seen to be N 2σ+4ε + 2ε. Hence we have I 22 = H τ + a n M 2 n uk σn u, u du + O N 2σ+4ε + O +2ε. By 4 and Stieltjes integration we may write this as I 22 = H τ + H N + τ N + M ym 2 y uk σy u, u dy du M 2 y uk σy u, u de y du + O N 2σ+4ε + O +2ε. If we estimate the second term as was done for the corresponding term in 4, we see that it also is N +θ 2σ+5ε + 2ε. In the first term we replace y by x + u. We then obtain 42 I 22 = H τ N u M x + um 2 xk σx, u dx du + O N +θ 2σ+5ε + O +2ε. Combining this with 4 in 39 we now find that I 2 = H τ + H N u τ M x + um 2 xk σx, u dx du N u M 2 x + um xk σ x, u dx du + ON +θ 2σ+5ε + O 2ε. Clearly I 3 is the complex conjugate of I 2, but with Bs instead of As

14 68 D. A. Goldston and S. M. Gonek and M x instead of M 2x. It therefore follows from 42 that 43 I 3 = H τ + H N u τ M 2 x + um xk σ x, u dx du N u M x + um 2 xk σx, u dx du + ON +θ 2σ+5ε + O 2ε, which is identical to the expression for I 2. Finally, we come to I 4. By 5 and 6 we see that 44 I 4 = N N x N N + M xm 2 yxy σ ΨU 2π log y x x = I 4 + I 42, M 2 xm yxy σ Ψ U 2π log y x dy dx dy dx say. In I 4 we set y = x + u and use 6 and 9 for u > x/τ to obtain I 4 = = N N minx/τ,n x Ψ U 2π log + u x minx/τ,n x M xm 2 x + ux 2σ + u σ x du dx + O M xm 2 x + uk σx, u du dx + O. Next we change the order of integration and find that I 4 = τ + H N u τ N u M xm 2 x + uk σx, u dx du By 6, 2, and 37a the first term is τ N M xm 2 x + uk σx, u dx du + O. x ε 2σ dx +ε N 2σ+2ε + N 2σ+4ε + ε.

15 hus, I 4 = H τ N u Mean value theorems 69 M xm 2 x + uk σx, u dx du + O N 2σ+4ε + O +ε. Since I 42 is I 4 with M and M 2 interchanged, we also have I 42 = hus we find that I 4 = H τ N u M 2xM x + uk σx, u dx du + O N 2σ+4ε + O +ε. H τ + H N u τ M x + um 2 xk σx, u dx du N u M 2 x + um xk σ x, u dx du + ON 2σ+4ε + O ε. On combining 34, 38, and 42, 43, and 45, we obtain I = Ψ U + h H + H H h H τ τ N u N u a n b n n 2σ N h hτ N h hτ M x, hk σ x, h dx M 2 x, hk σx, h dx M x + um 2 xk σx, u dx du M 2 x + um xk σ x, u dx du + ON 2σ+maxθ,ϕ+5ε + O 2ε. his agrees with 26 so the proof of heorem is complete. We now deduce Corollary from heorem. In the second term on the right in 46 we replace N h by N and H by N/τ. hen by A 4 and 2

16 7 D. A. Goldston and S. M. Gonek this results in a change of at most hhn ε/2 hτ 2σ + N 2σ h H + N/τ+<h N/τ h ε/2 N Nτ/τ+ x 2σ+ε/2 dx N ε/2 τ 2σ H 2 2σ+ε + + N 2σ H 2+ε + + τ 3 N 2 2σ+ε N ε/2 τ 2σ N/τ 2 2σ+ε + N 2σ N/τ 2+ε + τ 3 N 2 2σ+ε τ 2 N 2 2σ+2ε N 2 2σ+4ε since σ < and N. Hence the second term on the right-hand side of 46 equals N h N/τ hτ M x, hk σ x, h dx + O N 2 2σ+4ε. By 23 we may replace K σ x, h by x 2σ ΨU h 2πx with a total error of at most 2ε N ε/4 2ε N ε/4 N h N/τ hτ h N/τ x 2σ+ε/4 dx hτ 2σ+ε/2 + N 2σ+ε/2 2ε N ε/4 τ 2σ+ε/2 N/τ 2 2σ+ε + + τ N 2 2σ+ε/2 +3ε N ε/4 N 2 2σ+ε + τ 2 2σ+ε N 2 2σ+5ε. hus, the expression above equals N h N/τ hτ M x, hx 2σ ΨU h 2πx dx + O N 2 2σ+5ε. If we write v for h /2πx and then change the order of summation and integration, we get C /2πτ /2πN h 2πNv/ M h h 2πv, h 2σ Ψ U vv 2σ 2 dv + O N 2 2σ+5ε. Finally, by A 4, 8, and 9 if we extend the interval of integration to infinity we change our term by a negligible amount. hus, the second term

17 C /2πN h 2πNv/ Mean value theorems 7 on the right-hand side of 46 is h M h 2πv, h 2σ Ψ U vv 2σ 2 dv + O N 2 2σ+5ε. Similarly, we see that the third term on the right-hand side of 46 is h M 2 h 2πv, h 2σ Ψ U vv 2σ 2 dv C /2πN h 2πNv/ + O N 2 2σ+5ε. In much the same way we find that the fourth term on the right in 46 equals 47 N/τ τ N u M x + um 2 xx 2σ ΨU dx du 2πx + O N 2 2σ+5ε. Now by 7 and the mean value theorem of differential calculus we have 48 M x + u = M x + Ox + ε/2 τ for u/x τ. Hence, replacing M x+u by M x and using the estimates σ < and N, and 7 we change the above by at most hus, 47 equals N/τ τ N τ N/τ τ τ N/τ N τ τ N x ε 2σ dx du τ 2σ+2ε + N 2σ+2ε du y 2σ+2ε dy + τ N 2 2σ+2ε +2ε N 2 2σ+3ε + N 2 2σ+5ε. u M xm 2 xx 2σ ΨU dx du + O N 2 2σ+5ε. 2πx Substituting v for u /2πx and then changing the order of integration, we

18 72 D. A. Goldston and S. M. Gonek find that this equals C /2πτ /2πτN 2πNv/ τ M u u M 2 u 2σ du Ψ U vv 2σ 2 dv 2πv 2πv + O N 2 2σ+5ε. Now by 6, M i u 2πv N ε i =, 2 in the rectangle [, τ ] [/2πNτ, /2πτ]. Using this and 7, we find that if we begin the u integral at zero, the first term changes by /2πτN N ε /τ 2 2σ /2πτ /2πNτ Ψ U v v 2σ 2 dv N ε /τ 2 2σ /τ 2σ + /Nτ 2σ N ε τ + N 2σ N 2ε + N 2σ+2ε. Moreover, if we then extend the v integral to infinity, this changes our expression by a negligible amount because of 6 and 8. hus, 47 equals C 2πNv/ u u M M 2 u 2σ du Ψ U vv 2σ 2 dv 2πv 2πv /2πτN + O N 2 2σ+5ε + ON 2ε. reating the fifth term in 46 in exactly the same way, we find that it equals C 2πNv/ u u M M 2 u 2σ du Ψ U vv 2σ 2 dv 2πv 2πv Combining all our results, we now obtain I = Ψ U + C + C /2πN a n b n n 2σ h 2πNv/ /2πN h 2πNv/ 2πNv/ M /2πτN 2C M M 2 + O N 2 2σ+5ε + ON 2ε. h h 2πv, h 2σ Ψ U vv 2σ 2 dv h h 2πv, h 2σ Ψ U vv 2σ 2 dv u u M 2 u 2σ du 2πv 2πv Re Ψ U vv 2σ 2 dv

19 Mean value theorems 73 + O N 2 2σ+5ε + ON 2σ+maxθ,ϕ+5ε + ON 2ε, which is the same as 27. hus, the proof of Corollary is complete. 3. Proof of heorem 2 and its corollary. Multiplying out in 2, we have t I = Ψ U t A sb s dt Ψ U A s M 2 xx s dx dt or + t Ψ U B s M xx s dx dt N t Ψ U M xx s dx M 2 yy s dy dt, N 49 I = I I 2 I 3 + I 4. In I we multiply the two series and note by A and our assumption that σ > that the resulting double series is absolutely convergent. We may therefore integrate termwise. Using 5 and 6, we then find that or I = Ψ U a n b n n 2σ N<n + a n b m mn σ ΨU 2π log m n N<n n<m + b n a m mn σ Ψ U 2π log m, n N<n n<m 5 I = Ψ U N<n a n b n n 2σ + I 2 + I 3. Setting m = n + h in I2 and using A and 9 for h n/τ, we see that I2 = a n b n+h n 2σ + h σ ΨU n 2π log + h n N<n h<n/τ + O N 2σ = a n b n+h K σ n, h + O N 2σ, N<n h<n/τ say. Changing the order of summation, which is permissible by absolute N N

20 74 D. A. Goldston and S. M. Gonek convergence, and then splitting the sum over h at, we obtain I2 = a n b n+h K σ n, h + a n b n+h K σ n, h h maxn,hτ<n + O N 2σ. By A, 2, 37b, and 29 the second term is n 2σ+ε/42σ 2 <h hτ<n τ 2σ+ε2σ 2 2 2σ ε = τ ε N 2σ ε/2 N 2σ. <h hus, by 9 and Stieltjes integration we have I 2 = h maxn,hτ + h maxn,hτ M x, hk σ x, h dx he second term equals 5 E x, hk σ x, h maxn,hτ h h hτ<n hτ 2σ+ε/22σ 2 K σ x, h de x, h + O ε/2 N 2σ. maxn,hτ E x, h x K σx, h dx. We may replace E x, h here by Ox ϕ if we can show that h x η for all h and x > maxn, hτ. his condition will be met if τ η/ η. But this follows immediately from 28 and 29, so we find that 5 is ε maxn, hτ ϕ 2σ h H = ε N ϕ 2σ + ε h N/τ N/τ<h hτ ϕ 2σ ε τ N 2σ+ϕ N 2σ+ϕ+4ε by 2, 2, and 37b. Combining our estimates, we see that I 2 = h maxn,hτ + O ε/2 N 2σ. M x, hk σ x, h dx + O N 2σ+ϕ+4ε

21 reating I 3 in the same way, we obtain I3 = h maxn,hτ + O ε/2 N 2σ. Hence, by 5 we have Mean value theorems 75 M 2x, hk σ x, h dx + O N 2σ+ϕ+4ε 52 I = Ψ U N<n + a n b n n 2σ h maxn,hτ + h maxn,hτ M x, hk σ x, h dx M 2 x, hk σx, h dx or + ON 2σ+ϕ+4ε + O ε/2 N 2σ. Next consider I2. By 5 and absolute convergence we have I2 = M 2 xx σ ΨU 2π log x dx n N<n N = a n n n σ + M 2 xx σ ΨU 2π log x dx, n N<n n a n n σ N 53 I 2 = I 2 + I 22. In I 2 we write x = n + u and use A, 6, and 9 for u n/τ to obtain I 2 = N<n a n n σ n/τ + O N 2σ = N<n a n n/τ M 2 n + un + u σ ΨU 2π log + u du n M 2 n + uk σn, u du + O N 2σ, say. Changing the order of summation and integration by absolute convergence and then splitting the u integral at, we find that

22 76 D. A. Goldston and S. M. Gonek I 2 = + maxn,<n <n a n M 2 n + uk σn, u du a n M 2 n + uk σn, u du + O N 2σ. By A, 6, 2, 37b, and 29, the second term is <n n 2σ+ε/42σ 2 du τ 2σ+ε2σ 2 2 2σ ε = τ ε N 2σ ε/2 N 2σ. hus, by 4 and Stieltjes integration, we have 54 I 2 = he second term is + maxn, 2σ+ε/22σ 2 du M xm 2 x + uk σx, u dx du maxn, M 2 x + uk σx, u de x du + O ε/2 N 2σ. M 2 x + uk σx, ue x maxn, du maxn, M 2 x + uk σx, u + M 2 x, u x K σx, u E x dx du H ε maxn, θ 2σ+ε/2 du ε N/τ H N θ 2σ+ε/2 du + ε N/τ ε τ N 2σ+θ+ε N 2σ+θ+5ε θ 2σ+ε/2 du

23 Mean value theorems 77 by A 2, 2, 2, and 37a and 37b. hus we find that 55 I 2 = maxn, M xm 2 x + uk σx, u dx du + O N 2σ+θ+tε + O ε/2 N 2σ. In I 22 we set x = n u and obtain I 22 = N<n a n n σ n N By 9 for u > n/τ +, this equals N<n M 2 n un u σ ΨU 2π log u du. n minn N,n/τ+ a n n σ M 2 n un u σ ΨU 2π log u du n = N<n a n minn N,n/τ+ + O N 2σ M 2 n uk σn u, u du + O N 2σ. As in I 2 we interchange the order of summation and integration and split the resulting integral at to obtain I 22 = + maxn+u,+<n +<n a n M 2 n uk σn u, u du a n M 2 n uk σn u, u du + O N 2σ. Estimating the second term as we did in the case of I 2, we see that it is ε/2 N 2σ. hus, by 4 and Stieltjes integration I 22 = + maxn+u,+ maxn+u,+ + O ε/2 N 2σ. M ym 2 y uk σy u, u dy du M 2 y uk σy u, u de y du he second term is estimated just like the corresponding term in 54 with the result that it also is N 2σ+θ+5ε. In the first term we make the

24 78 D. A. Goldston and S. M. Gonek substitution x = y u and find that I 22 = maxn, M 2 xm x + uk σ x, u dx du + O N 2σ+θ+5ε + O ε/2 N 2σ. Combining 53, 55, and 56, we now have I 2 = + maxn, maxn, M x + um 2 xk σx, u dx du M 2 x + um xk σ x, u dx du + ON 2σ+θ+5ε + O ε/2 N 2σ. Since I 3 is the complex conjugate of I 2 with B s replacing A s and M x replacing M 2x, it follows from 57 that 58 I 3 = + maxn, maxn, M 2 x + um xk σ x, u dx du M x + um 2 xk σx, u dx du + ON 2σ+θ+5ε + O ε/2 N 2σ. Note that this is identical to the expression for I2. Next we treat I4. By absolute convergence, 5, and 6 we have I4 = M xm 2 yxy σ ΨU 2π log y dy dx x N + x N x = I 4 + I 42, M 2 xm yxy σ Ψ U 2π log y x dy dx say. In I 4 we set y = x + u and use 6 and 9 for u > x/τ to obtain I 4 = N x/τ M xm 2 x + ux 2σ + u σ ΨU x 2π log + u du dx x + O N 2σ

25 = N x/τ Mean value theorems 79 M xm 2 x + uk σx, u du dx + O N 2σ. he double integral converges absolutely so we may change the order of integration. After doing so and splitting the resulting u integral at, we obtain I 4 = + maxn, M xm 2 x + uk σx, u dx du M xm 2 x + uk σx, u dx du + O N 2σ. By 6, 2, 37a, and 29, the second term is x 2σ+ε/42σ 2 dx du τ 2σ+ε2σ 2 2 2σ ε τ ε N 2σ ε/2 N 2σ. hus we find that I 4 = maxn, 2σ+ε/22σ 2 du M xm 2 x + uk σx, u dx du + O ε/2 N 2σ. Since I 42 is I 4 with M and M 2 interchanged, we see that I 42 = maxn, hus, we find that 59 I 4 = + M 2xM x + uk σx, u dx du + O ε/2 N 2σ. maxn, maxn, + O ε/2 N 2σ. M x + um 2 xk σx, u dx du M 2 x + um xk σ x, u dx du By 49, 52, 57, 58, and 59, we now see that

26 8 D. A. Goldston and S. M. Gonek 6 I = Ψ U N<n + + a n b n n 2σ h maxn,hτ h maxn,hτ maxn, maxn, M x, hk σ x, h dx M 2 x, hk σx, h dx M x + um 2 xk σx, u dx du M 2 x + um xk σ x, u dx du + ON 2σ+maxθ,ϕ+5ε + O ε/2 N 2σ. his is 3 so the proof of heorem 2 is complete. he proof of Corollary 2 is along the same lines as that of Corollary so we leave out most of the details. Replacing K σ x, u by x 2σ u ΨU 2πx and M i x + u by M i x i =, 2, we see from 23, 48, and the fact that σ > and N, that the right-hand side of 6 changes by no more than O N 2 2σ+5ε. herefore we have 6 I = Ψ U N<n a n b n n 2σ h maxn,hτ h maxn,hτ maxn, h M x, hx 2σ ΨU dx 2πx h M 2 x, hx 2σ ΨU dx 2πx M xm 2 xx 2σ Re Ψ U u 2πx + ON 2σ+maxθ,ϕ+5ε + O ε/2 N 2σ + O N 2 2σ+5ε. N/τ<h N M x, hx 2σ ΨU h 2πx dx du Next consider the second term on the right-hand side of 6. If we replace the lower limit of integration by N, then this changes the term by the amount hτ dx.

27 h N Mean value theorems 8 For this range of h and x we have h/2πx τ, so by 9 this term is negligible. We may therefore take the second term to be dx. M x, hx 2σ ΨU h 2πx We now set v = h/2πx and find that this equals h /2πN C h 2σ M h h 2πv, h v 2σ 2 ΨU v dv. Changing the order of summation and integration, we find that this is /2πN h C h 2πv, h 2σ Ψ U vv 2σ 2 dv + C H h M /2πN /2πN 2πNv/ <h M Similarly, the third term on the right-hand side of 6 is C /2πN + C H h M 2 /2πN /2πN h h 2πv, h 2σ Ψ U vv 2σ 2 dv 2πNv/ <h M 2 h h 2πv, h 2σ Ψ U vv 2σ 2 dv. h h 2πv, h 2σ Ψ U vv 2σ 2 dv. Finally, the same basic analysis applied to the fourth term on the right-hand side of 6 leads to 2C H /2πN H 2πNv/ M u u M 2 u 2σ du Re 2πv 2πv Ψ U vv 2σ 2 dv. Combining our expressions, we obtain 3, so the proof of Corollary 2 is complete. 4. Proof of heorem 3. We have t J = Ψ U As N M xx s dx B s N M 2 dx dt xx s

28 82 D. A. Goldston and S. M. Gonek = + Ψ U t Ψ U t AsB s dt As N N M 2 xx s dx dt t Ψ U B s t Ψ U N M xx s dx = J J 2 J 3 + J 4, say. By 5 we see that J = N<m M xx s dx dt N M 2 xx s dx dt a n b m n σ m σ ΨU 2π log m n Now if either m N/N or N n/n is greater than τ, then logm/n τ so that by 9, Ψ U 2π log m D 2 for any D 2. n Hence, we have J = N τ N<m N+τ + O N 2 σ σ, say. By A, 7, and 37b, this is N /τ N<m N+/τ τ 2 N 2 σ σ +ε + N 2 σ σ +2ε N 2 σ σ +ε N 2 σ σ +5ε.. a n b m n σ m σ ΨU 2π log m n n ε/2 σ m ε/2 σ + N 2 σ σ he integrals J 2, J 3, and J 4 are treated similarly with the same result. hus, J N 2 σ σ +5ε and the proof of the theorem is complete.

29 Mean value theorems Four examples. he following examples illustrate the application of some of our results. Example. Let a n = b n = for n =, 2,... hen so that we may take and Ax = Bx = C x, h = C 2 x, h = [x] M x = M 2 x = M x, h = M 2 x, h = x E x = E 2 x = E x, h = E 2 x, h. In Corollary we may therefore take θ = ϕ = and η = ε, where < ε < /2 is arbitrarily small. Also, taking U = log, N, and σ = /2, we find that t I = Ψ U n /2 it x /2 it dx 2 dt = Ψ U [ ] 2πN n + 2 v Re Ψ U vv dv 2 /2πNτ + O N +5ε. /2πN N 2πN v Re Ψ U vv dv Since Ψ U v, the lower limit of integration in the third term may be replaced by /2πN with an error of O. hus we may rewrite the above as I = Ψ U [ ] 2πN 2πN log N 2 v v Re 2 Ψ U vv dv + 2 /2πN [ 2πN /2πN Re Ψ U vv dv ] v + O + O N +5ε. For v we have 2πN v Ψ U vv dv

30 84 D. A. Goldston and S. M. Gonek 62 ΨU v = Ψ U + OC 2 C + U v and Ψ U is real. hus, the second term on the right-hand side above is 2 Ψ U /2πN he third term equals Ψ U /2πN 2πN = 2 Ψ U = O. Finally, by 7 the fourth term is hus we find that U [ ] 2πN v v v dv + O 2 2πN/ y [y] /2y dy + O v dv + O = Ψ U log/n + O. v dv + U U v 2 dv log log. I = Ψ U + o log + O N +5ε. If in the definition of Ψ U t we take C = U and C 2 = U and Ψ U t, then Ψ U t is a minorant for the characteristic function of the interval [,]. On the other hand, taking C = U and C 2 = + U, we obtain a majorant. Since in either case it follows that 63 Ψ U = + OU, n /2 it N x /2 it dx 2 dt log for N 2/+5ε. Notice that if N, then the mean-square of N x /2 it dx is N, so by the left-hand side of 63 is log N. We conclude this example by remarking that since θ = ϕ = and η = ε, a straightforward but tedious application of heorem would allow us to prove that 63 is in fact valid for N A for any fixed A. It is interesting to note that this and the simple approximation see itchmarsh [5; p. 49]

31 ζ 2 + it = n /2 it N Mean value theorems 85 x /2 it dx + O 2 + it N /2 + O 2 + it with N = 2, gives the classical mean value formula 2dt ζ 2 + it log. Example 2. Let a n = enα, b n = enβ for n =, 2,..., with < α, β <. Consider first the case where α = β. hen Ax = Bx = n x enα,, so we may take M i x = and E i x i =, 2. Also, C i x, h = e hα[x], so we may take M i x, h = e hαx and E i x, h i =, 2. hus θ = ϕ = and we may take η = ε. aking U = log 2, N, and σ = /2 in Corollary, we find that t I = Ψ U enαn /2 it 2 dt = Ψ U n + 2 Re + O N +5ε /2πN = Ψ U log N + γ + O/N [ e α 2πN 2 Re /2πN + O N +5ε. By 62 the middle term equals 2 Ψ U Re eα v] eα /2πN h 2πNv/ Ψ U vv dv e hα ΨU vv dv [ ] 2πN e α v v dv + OC 2 C + U + O Ψ U v v dv.

32 86 D. A. Goldston and S. M. Gonek he second error term here is C 2 C + U U C 2 C + U log log v dv + U U v 2 dv by 7. hus, changing variables in the remaining integral, we find that the middle term above equals 2πN/ 2 Ψ U Re e α[y] y dy eα Now 2πN/ = e α[y] y dy [2πN/ ] k= k+ e αk = log/n + O k y dy + e α + OC 2 C + U log log. [ 2πN ] log 2πN/ 2πN log [2πN/ ] by partial summation. Hence, our middle term is equal to 2 Ψ U Re logn/ + OC 2 C + U log log. eα Observing that Reeα = /2 and combining our results, we find that 64 I = Ψ U log + OC 2 C + U log log + O N +5ε. If we remove the weight function Ψ U t/ as in the last example, we deduce that 65 enαn /2 it 2 dt log for N 2/+5ε. Note that by the left-hand side is log N when N <. Had we used heorem rather than Corollary, we could have shown with more work that 65 in fact holds for N A for any A. Now consider the case when α β. Here we may take M x = M 2 x = M x, h = M 2 x, h =

33 and Mean value theorems 87 E x, E 2 x, E x, h, E 2 x, h. It is particularly easy to use heorem in such a case. We take θ = ϕ =, η = ε, σ = /2, U = log 2, and ε N ε/5ε, and find that t I = Ψ U enαn /2 it e nβn /2+it dt = Ψ U enα βn + ON 5ε = Ψ U log eα β + O ε. If we are interested in the unweighted integral I = enαn /2 it e nβn /2+it dt instead, we can proceed as follows. We take C = U and C 2 = + U in the definition of Ψ U in I and use the Cauchy Schwarz inequality to see that { +2U I I + enαn /2 it 2 } /2 dt 2U { +2U enβn /2 it 2 } /2. dt 2U + hese integrals can be estimated by using 64 with C = 3U, C 2 = 3U in the definition of Ψ U and then with C = 3U, C 2 = + 3U. his gives I I U log = o provided that N 2 ε/+5ε. hus, for α β and for the same range of N, we have 66 enαn /2 it e nβn /2+it dt = + o log eα β. Had we proved a longer-range version of 64 by appealing to heorem instead of Corollary, 66 would also hold in such a range.

### Statistics of the Zeta zeros: Mesoscopic and macroscopic phenomena

Statistics of the Zeta zeros: Mesoscopic and macroscopic phenomena Department of Mathematics UCLA Fall 2012 he Riemann Zeta function Non-trivial zeros: those with real part in (0, 1). First few: 1 2 +

### 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

### Counting Primes whose Sum of Digits is Prime

2 3 47 6 23 Journal of Integer Sequences, Vol. 5 (202), Article 2.2.2 Counting Primes whose Sum of Digits is Prime Glyn Harman Department of Mathematics Royal Holloway, University of London Egham Surrey

### Lectures 5-6: Taylor Series

Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

### 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

### 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:

CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### 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

### Recent progress in additive prime number theory

Recent progress in additive prime number theory University of California, Los Angeles Mahler Lecture Series Additive prime number theory Additive prime number theory is the study of additive patterns in

### Differentiating under an integral sign

CALIFORNIA INSTITUTE OF TECHNOLOGY Ma 2b KC Border Introduction to Probability and Statistics February 213 Differentiating under an integral sign In the derivation of Maximum Likelihood Estimators, or

### a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

### 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

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### 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,

### Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.

Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,

### 5 Indefinite integral

5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse

### THE CENTRAL LIMIT THEOREM TORONTO

THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### Doug Ravenel. October 15, 2008

Doug Ravenel University of Rochester October 15, 2008 s about Euclid s Some s about primes that every mathematician should know (Euclid, 300 BC) There are infinitely numbers. is very elementary, and we

### 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 ) =

### 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

### The Epsilon-Delta Limit Definition:

The Epsilon-Delta Limit Definition: A Few Examples Nick Rauh 1. Prove that lim x a x 2 = a 2. (Since we leave a arbitrary, this is the same as showing x 2 is continuous.) Proof: Let > 0. We wish to find

### An example of a computable

An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

### Jacobi s four squares identity Martin Klazar

Jacobi s four squares identity Martin Klazar (lecture on the 7-th PhD conference) Ostrava, September 10, 013 C. Jacobi [] in 189 proved that for any integer n 1, r (n) = #{(x 1, x, x 3, x ) Z ( i=1 x i

### N E W S A N D L E T T E R S

N E W S A N D L E T T E R S 73rd Annual William Lowell Putnam Mathematical Competition Editor s Note: Additional solutions will be printed in the Monthly later in the year. PROBLEMS A1. Let d 1, d,...,

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### 8.7 Mathematical Induction

8.7. MATHEMATICAL INDUCTION 8-135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture

### 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

### 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

### LIMITATIONS TO MOLLIFYING ζ(s).

LIMIAIONS O MOLLIFYING ζs. MAKSYM RADZIWI L L Abstract. We establish limitations to how well one can mollify ζs on the critical line with mollifiers of arbitrary length. Our result gives a non-trivial

### SECTION 10-2 Mathematical Induction

73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

### 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,

### 1 The Dirichlet Problem. 2 The Poisson kernel. Math 857 Fall 2015

Math 857 Fall 2015 1 The Dirichlet Problem Before continuing to Fourier integrals, we consider first an application of Fourier series. Let Ω R 2 be open and connected (region). Recall from complex analysis

### 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

### 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

### Polynomials Classwork

Polynomials Classwork What Is a Polynomial Function? Numerical, Analytical and Graphical Approaches Anatomy of an n th -degree polynomial function Def.: A polynomial function of degree n in the vaiable

### Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

### Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.

M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### Section 6-2 Mathematical Induction

6- Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that

### MATH2001 Development of Mathematical Ideas History of Solving Polynomial Equations

MATH2001 Development of Mathematical Ideas History of Solving Polynomial Equations 19/24 April 2012 Lagrange s work on general solution formulae for polynomial equations The formulae for the cubic and

### Linear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:

Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrix-vector notation as 4 {{ A x x x {{ x = 6 {{ b General

### Fourier Series. A Fourier series is an infinite series of the form. a + b n cos(nωx) +

Fourier Series A Fourier series is an infinite series of the form a b n cos(nωx) c n sin(nωx). Virtually any periodic function that arises in applications can be represented as the sum of a Fourier series.

### Chapter 7. Induction and Recursion. Part 1. Mathematical Induction

Chapter 7. Induction and Recursion Part 1. Mathematical Induction The principle of mathematical induction is this: to establish an infinite sequence of propositions P 1, P 2, P 3,..., P n,... (or, simply

### 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

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

### Algebraic and Transcendental Numbers

Pondicherry University July 2000 Algebraic and Transcendental Numbers Stéphane Fischler This text is meant to be an introduction to algebraic and transcendental numbers. For a detailed (though elementary)

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### On the Union of Arithmetic Progressions

On the Union of Arithmetic Progressions Shoni Gilboa Rom Pinchasi August, 04 Abstract We show that for any integer n and real ɛ > 0, the union of n arithmetic progressions with pairwise distinct differences,

### CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### STRUCTURE AND RANDOMNESS IN THE PRIME NUMBERS. 1. Introduction. The prime numbers 2, 3, 5, 7,... are one of the oldest topics studied in mathematics.

STRUCTURE AND RANDOMNESS IN THE PRIME NUMBERS TERENCE TAO Abstract. A quick tour through some topics in analytic prime number theory.. Introduction The prime numbers 2, 3, 5, 7,... are one of the oldest

### Double Sequences and Double Series

Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine E-mail: habil@iugaza.edu Abstract This research considers two traditional important questions,

### CHAPTER 3. Sequences. 1. Basic Properties

CHAPTER 3 Sequences We begin our study of analysis with sequences. There are several reasons for starting here. First, sequences are the simplest way to introduce limits, the central idea of calculus.

### 1 Lecture: Integration of rational functions by decomposition

Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

### (Refer Slide Time: 01:11-01:27)

Digital Signal Processing Prof. S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 6 Digital systems (contd.); inverse systems, stability, FIR and IIR,

### The sum of digits of polynomial values in arithmetic progressions

The sum of digits of polynomial values in arithmetic progressions Thomas Stoll Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France E-mail: stoll@iml.univ-mrs.fr

### Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients

DOI: 10.2478/auom-2014-0007 An. Şt. Univ. Ovidius Constanţa Vol. 221),2014, 73 84 Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients Anca

### CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS

CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS JEREMY BOOHER Continued fractions usually get short-changed at PROMYS, but they are interesting in their own right and useful in other areas

### Computing divisors and common multiples of quasi-linear ordinary differential equations

Computing divisors and common multiples of quasi-linear ordinary differential equations Dima Grigoriev CNRS, Mathématiques, Université de Lille Villeneuve d Ascq, 59655, France Dmitry.Grigoryev@math.univ-lille1.fr

### Course Notes for Math 162: Mathematical Statistics Approximation Methods in Statistics

Course Notes for Math 16: Mathematical Statistics Approximation Methods in Statistics Adam Merberg and Steven J. Miller August 18, 6 Abstract We introduce some of the approximation methods commonly used

### The Relation between Two Present Value Formulae

James Ciecka, Gary Skoog, and Gerald Martin. 009. The Relation between Two Present Value Formulae. Journal of Legal Economics 15(): pp. 61-74. The Relation between Two Present Value Formulae James E. Ciecka,

### MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 4: Fourier Series and L 2 ([ π, π], µ) ( 1 π

MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 4: Fourier Series and L ([, π], µ) Square Integrable Functions Definition. Let f : [, π] R be measurable. We say that f

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of Rou-Huai Wang

A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of Rou-Huai Wang 1. Introduction In this note we consider semistable

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

### CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

### NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

### 4.3 Limit of a Sequence: Theorems

4.3. LIMIT OF A SEQUENCE: THEOREMS 5 4.3 Limit of a Sequence: Theorems These theorems fall in two categories. The first category deals with ways to combine sequences. Like numbers, sequences can be added,

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### Taylor and Maclaurin Series

Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

### 5. Convergence of sequences of random variables

5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,

### a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2

Solving Quadratic Equations By Square Root Method Solving Quadratic Equations By Completing The Square Consider the equation x = a, which we now solve: x = a x a = 0 (x a)(x + a) = 0 x a = 0 x + a = 0

### Prime and Composite Terms in Sloane s Sequence A056542

1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.3 Prime and Composite Terms in Sloane s Sequence A056542 Tom Müller Institute for Cusanus-Research University and Theological

### The Characteristic Polynomial

Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

### 20 Applications of Fourier transform to differential equations

20 Applications of Fourier transform to differential equations Now I did all the preparatory work to be able to apply the Fourier transform to differential equations. The key property that is at use here

### 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

### FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

### SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION

CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Alessandro Artale UniBZ - http://www.inf.unibz.it/ artale/ SECTION 5.2 Mathematical Induction I Copyright Cengage Learning. All rights reserved.

### On the largest prime factor of x 2 1

On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical

### ARITHMETICAL FUNCTIONS II: CONVOLUTION AND INVERSION

ARITHMETICAL FUNCTIONS II: CONVOLUTION AND INVERSION PETE L. CLARK 1. Sums over divisors, convolution and Möbius Inversion The proof of the multiplicativity of the functions σ k, easy though it was, actually

### On the Least Prime Number in a Beatty Sequence. Jörn Steuding (Würzburg) - joint work with Marc Technau - Salamanca, 31 July 2015

On the Least Prime Number in a Beatty Sequence Jörn Steuding (Würzburg) - joint work with Marc Technau - Salamanca, 31 July 2015 i. Beatty Sequences ii. Prime Numbers in a Beatty Sequence iii. The Least

### Analysis MA131. University of Warwick. Term

Analysis MA131 University of Warwick Term 1 01 13 September 8, 01 Contents 1 Inequalities 5 1.1 What are Inequalities?........................ 5 1. Using Graphs............................. 6 1.3 Case

### Limits and convergence.

Chapter 2 Limits and convergence. 2.1 Limit points of a set of real numbers 2.1.1 Limit points of a set. DEFINITION: A point x R is a limit point of a set E R if for all ε > 0 the set (x ε,x + ε) E is

### Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree

### Sample Induction Proofs

Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given

### The Black-Scholes-Merton Approach to Pricing Options

he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining

### Concentration of points on two and three dimensional modular hyperbolas and applications

Concentration of points on two and three dimensional modular hyperbolas and applications J. Cilleruelo Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas Universidad

### 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