Chapter 7 The Prime Number Theorem


 Grant Fitzgerald
 3 years ago
 Views:
Transcription
1 Chapter 7 The Prime Number Theorem In this final chapter we will take advantage of an opportunity to apply many of the ideas and results from earlier chapters in order to give an analytic proof of the famous prime number theorem: If π(x) is the number of primes less than or equal to x, then x π(x)lnx asx.that is, π(x) is asymptotically equal to x/ ln x as x. (In the sequel, prime will be taken to mean positive prime.) Perhaps the first recorded property of π(x) is that π(x) as x, in other words, the number of primes is infinite.this appears in Euclid s Elements.A more precise result that was established much later by Euler (737) is that the series of reciprocals of the prime numbers, , is a divergent series.this can be interpreted in a certain sense as a statement about how fast π(x) as x.later, near the end of the 8th century, mathematicians, including Gauss and Legendre, through mainly empirical considerations, put forth conectures that are equivalent to the above statement of the prime number theorem (PNT).However, it was not until nearly years later, after much effort by numerous 9th century mathematicians, that the theorem was finally established (independently) by Hadamard and de la Vallée Poussin in 896.The quest for a proof led Riemann, for example, to develop complex variable methods to attack the PNT and related questions. In the process, he made a remarkable and as yet unresolved conecture known as the Riemann hypothesis, whose precise statement will be given later.now it is not clear on the surface that there is a connection between complex analysis and the distribution of prime numbers.but in fact, every proof of the PNT dating from Hadamard and de la Vallée Poussin, up to 949 when P.Erdös and A.Selberg succeeded in finding elementary proofs, has used the methods of complex variables in an essential way.in 98, D.J. Newman published a new proof of the PNT which, although still using complex analysis, nevertheless represents a significant simplification of previous proofs.it is Newman s proof, as modified by J.Korevaar, that we present in this chapter. There are a number of preliminaries that must be dealt with before Newman s method can be applied to produce the theorem.the proof remains far from trivial but the steps
2 2 CHAPTER 7. THE PRIME NUMBER THEOREM along the way are of great interest and importance in themselves.we begin by introducing the Riemann zeta function, which arises via Euler s product formula and forms a key link between the sequence of prime numbers and the methods of complex variables. 7. The Riemann Zeta function The Riemann zeta function is defined by ζ(z) = where n z = e z ln n.since n z = n Re z, the given series converges absolutely on Re z> and uniformly on {z :Rez +δ} for every δ>.let p,p 2,p 3,... be the sequence 2,3,5,... of prime numbers and note that for =, 2,... and Re z>, we have n= /p z =+ p z Now consider the partial product m m p z = + = =( p z n z + p 2z +. + p 2z + ). By multiplying the finitely many absolutely convergent series on the right together, rearranging, and applying the fundamental theorem of arithmetic, we find that the product is the same as the sum n P m n, where P z m consists of along with those positive integers whose prime factorization uses only primes from the set {p,...,p m }.Therefore We now state this formally. = p z 7.. Euler s Product formula = n=, Re z>. nz For Re z>, the Riemann zeta function ζ(z) = n= /nz is given by the product ( ) p z = where {p } is the (increasing) sequence of prime numbers. The above series and product converge uniformly on compact subsets of Re z>, hence ζ is analytic on Re z >.Furthermore, the product representation of ζ shows that ζ has no zeros in Re z> (Theorem 6..7). Our proof of the PNT requires a number of additional properties of ζ.the first result is concerned with extending ζ to a region larger than Re z>.
3 7.. THE RIEMANN ZETA FUNCTION Extension Theorem for Zeta The function ζ(z) /(z ) has an analytic extension to the right half plane Re z>. Thus ζ has an analytic extension to {z :Rez>,z } and has a simple pole with residue at z =. Proof.For Re z>, apply the summation by parts formula (Problem 2.2.7) with a n = n and b n =/n z to obtain Thus But k n= [ n (n +) z n z ] k + (n +) z = n= [ n (n +) z ] n z = nz = k k z (n +) z. k k z n+ n n= [ n n= t z dt = z (n +) z n z n+ where [t] is the largest integer less than or equal to t.hence we have k n= k n z =+ (n +) z = k k z + z n= Letting k, we obtain the integral formula ζ(z) =z = k z + z n= k for Re z>.consider, however, the closely related integral z tt z dt = z Combining this with () we can write ζ(z) z =+z n n+ n [t]t z dt. ]. [t]t z dt [t]t z dt [t]t z dt () t z dt = z z =+ z. ([t] t)t z dt. Now fix k> and consider the integral k ([t] t)t z dt. By (3.3.3), this integral is an entire function of z.furthermore, if Re z>, then k ([t] t)t z dt k t Re(z+) dt t Re z dt = Re z.
4 4 CHAPTER 7. THE PRIME NUMBER THEOREM This implies that the sequence f k (z) = k ([t] t)t z dt of analytic functions on Re z> is uniformly bounded on compact subsets. Hence by Vitali s theorem (5..4), the limit function f(z) = ([t] t)t z dt (as the uniform limit on compact subsets of Re z>) is analytic, and thus the function +z ([t] t)t z dt is also analytic on Re z>.but this function agrees with ζ(z) z for Re z>, and consequently provides the required analytic extension of ζ to Re z >.This completes the proof of the theorem. We have seen that Euler s formula (7..) implies that ζ has no zeros in the half plane Re z>, but how about zeros of (the extension of) ζ in < Re z? The next theorem asserts that ζ has no zeros on the line Re z =.This fact is crucial to our proof of the PNT. 7..3Theorem The Riemann zeta function has no zeros on Re z =,so(z )ζ(z) is analytic and zerofree on a neighborhood of Re z. Proof.Fix a real number y and consider the auxiliary function = h(x) =ζ 3 (x)ζ 4 (x + iy)ζ(x + i2y) for x real and x>.by Euler s product formula, if Re z> then ln ζ(z) = ln p z = Re Log( p z )=Re = = n= n p nz where we have used the expansion Log( w) = n= wn /n, valid for w <.Hence ln h(x) =3ln ζ(x) +4ln ζ(x + iy) +ln ζ(x + i2y) =3Re = = n= +Re = n= n p nx n p nx = n= But p iny = e iny and p i2ny form +4Re n p nx i2ny Re(3 + 4p iny = n= n p nx iny + p i2ny ). = e i2ny.thus Re(3 + 4p iny 3+4cosθ + cos 2θ =3+4cosθ + 2 cos 2 θ = 2( + cos θ) 2. + p i2ny ) has the
5 7.. THE RIEMANN ZETA FUNCTION 5 Therefore ln h(x) and consequently h(x) = ζ 3 (x) ζ 4 (x + iy) ζ(x + i2y). Thus h(x) x = (x )ζ(x) 3 ζ(x + iy) x 4 ζ(x + i2y) x. But if ζ(+iy) =, then the left hand side of this inequality would approach a finite limit ζ ( + iy) 4 ζ( + i2y) as x + since ζ has a simple pole at with residue.however, the right hand side of the inequality contradicts this.we conclude that ζ( + iy). Since y is an arbitrary nonzero real number, ζ has no zeros on Re z =. Remark The ingenious introduction of the auxiliary function h is due to Mertens (898).We now have shown that any zeros of ζ in Re z > must lie in the strip < Re z <. The study of the zeros of ζ has long been the subect of intensive investigation by many mathematicians.riemann had stated in his seminal 859 paper that he considered it very likely that all the zeros of ζ in the above strip, called the critical strip, lie on the line Re z = /2.This assertion is now known as the Riemann hypothesis, and remains as yet unresolved.however, a great deal is known about the distribution of the zeros of ζ in the critical strip, and the subect continues to capture the attention of eminent mathematicians.to state ust one such result, G.H.Hardy proved in 95 that ζ has infinitely many zeros on the line Re z = /2.Those interested in learning more about this fascinating subect may consult, for example, the book Riemann s Zeta Function by H.M. Edwards. Another source is We turn next to zeta s logarithmic derivative ζ /ζ, which we know is analytic on Re z >. In fact, more is true, for by (7..3), ζ /ζ is analytic on a neighborhood of {z :Rez and z }.Since ζ has a simple pole at z =,sodoesζ /ζ, with residue Res(ζ /ζ, ) =. [See the proof of (4.2.7).] We next obtain an integral representation for ζ /ζ that is similar to the representation () above for ζ. [See the proof of (7..2).] But first, we must introduce the von Mangoldt function Λ, which is defined by { if n = p m for some m, Λ(n) = otherwise. Thus Λ(n) islnp if n is a power of the prime p, and is if not.next define ψ on x by ψ(x) = n x Λ(n). (2) An equivalent expression for ψ is ψ(x) = p x m p (x)lnp,
6 6 CHAPTER 7. THE PRIME NUMBER THEOREM where the sum is over primes p x and m p (x) is the largest integer such that p mp(x) x. (For example, ψ(.4) = 3 ln ln [ 3 + ln ] 5 + ln 7.) Note that p mp(x) x iff m p (x)lnp ln x iff m p (x) ln x.thus m ln x p(x) = where as before, [ ] denotes the greatest integer function.the function ψ will be used to obtain the desired integral representation for ζ /ζ Theorem For Re z>, ζ (z) ζ(z) = z ψ(t)t z dt (3) where ψ is defined as above. Proof.In the formulas below, p and q range over primes.if Re z>, we have ζ(z) = p ( p z ) by (7..), hence ζ (z) = p = ζ(z) p = ζ(z) p p z ( p z ) 2 q z q p p z ( p z ) 2 ( p z ) p z p z. Thus ζ (z) ζ(z) = p p z p z = p p nz. n= The iterated sum is absolutely convergent for Re z>, so it can be rearranged as a double sum (p,n),n (p n ) z = k z k where k = p n for some n.consequently, ζ (z) ζ(z) = k z Λ(k) = k z (ψ(k) ψ(k )) k= by the definitions of Λ and ψ.but using partial summation once again we obtain, with a k = k z,b k+ = ψ(k), and b = ψ() = in Problem 2.2.7, k= k= M M k z (ψ(k) ψ(k )) = ψ(m)(m +) z + ψ(k)(k z (k +) z ). k=
7 7.2. AN EQUIVALENT VERSION OF THE PRIME NUMBER THEOREM 7 Now from the definition (2) of ψ(x) wehaveψ(x) x ln x, so if Rez > wehave ψ(m)(m +) z asm.moreover, we can write M M ψ(k)(k z (k +) z )= ψ(k)z k= = = z k= M z k= k M k+ k+ k t z dt ψ(t)t z dt ψ(t)t z dt because ψ is constant on each interval [k, k + ).Taking limits as M, we finally get ζ (z) ζ(z) = z ψ(t)t z dt, Re z>. 7.2 An Equivalent Version of the Prime Number Theorem The function ψ defined in (2) above provides yet another connection, through (3), between the Riemann zeta function and properties of the prime numbers.the integral that appears in (3) is called the Mellin transform of ψ and is studied in its own right.we next establish a reduction, due to Chebyshev, of the prime number theorem to a statement involving the function ψ Theorem The prime number theorem holds, that is, x π(x)lnx, iff x ψ(x) asx. Proof.Recall that ψ(x) = [ ] ln x p x ln x () p x =lnx p x = (ln x)π(x).
8 8 CHAPTER 7. THE PRIME NUMBER THEOREM However, if <y<x, then π(x) =π(y)+ π(y)+ <y+ ln y y<p x y<p x y<p x y + ln y ψ(x). ln y (2) Now take y = x/(ln x) 2 in (2), and we get π(x) x (ln x) 2 + ln x 2lnlnx ψ(x). Thus π(x) ln x x ln x + ln x ψ(x) ln x 2lnlnx x. (3) It now follows from () and (3) that ψ(x) x ln x x π(x) ln x + ln x ψ(x) ln x 2lnlnx x and from this we can see that x ψ(x) iffx π(x)lnx asx. The goal will now be to show that ψ(x)/x asx.a necessary intermediate step for our proof is to establish the following weaker estimate on the asymptotic behavior of ψ(x) Lemma There exists C> such that ψ(x) Cx, x >.For short, ψ(x) =O(x). Proof.Again recall that ψ(x) = p x [ ln x ]lnp, x >.Fix x> and let m be an integer such that 2 m <x 2 m+.then ψ(x) =ψ(2 m )+ψ(x) ψ(2 m ) ψ(2 m )+ψ(2 m+ ) ψ(2 m ) = [ ] ln 2 m + p 2 m Consider, for any positive integer n, n<p 2n =ln 2 m <p 2 m+ n<p 2n p. [ ln 2 m+ ].
9 7.3. PROOF OF THE PRIME NUMBER THEOREM 9 Now for any prime p such that n<p 2n, p divides (2n)!/n! =n! ( ) 2n n.since such a p does not divide n!, it follows that p divides ( ) 2n n.hence ( ) 2n p < (+) 2n =2 2n, n n<p 2n and we arrive at n<p 2n <2n ln 2. Therefore m = m < 2 k ln 2 < 2 m+ ln 2 p 2 m k= 2 k <p 2 k k= and <2 m+ ln 2. 2 m <p 2 m+ [ ] ln x But if p x is such that >, then ln x 2 and hence x p2 so that x p. Thus those terms in the sum [ ] [ ] ln x ln x p x where > occur only when p x, and the sum of terms of this form contribute no more than ln x = π( x)lnx. p x It follows from the above discussion that if 2 m <x 2 m+, then ψ(x) 2 m+ ln2+2 m+ ln 2 + π( x)lnx =2 m+2 ln 2 + π( x)lnx < 4x ln 2 + π( x)lnx 4x ln 2 + x ln x =(4ln2+ ln x)x. x Since x ln x asx, we conclude that ψ(x) =O(x), which proves the lemma. 7.3Proof of the Prime Number Theorem Our approach to the prime number theorem has been along traditional lines, but at this stage we will apply D.J. Newman s method (Simple Analytic Proof of the Prime Number Theorem, American Math.Monthly 87 (98), ) as modified by J.Korevaar (On
10 CHAPTER 7. THE PRIME NUMBER THEOREM Newman s Quick Way to the Prime Number Theorem, Math.Intelligencer 4 (982), 85).Korevaar s approach is to apply Newman s ideas to obtain properties of certain Laplace integrals that lead to the prime number theorem. Our plan is to deduce the prime number theorem from a Tauberian theorem (7.3.) and its corollary (7.3.2). Then we will prove (7.3.) and (7.3.2) Auxiliary Tauberian Theorem Let F be bounded and piecewise continuous on [, + ), so that its Laplace transform G(z) = F (t)e zt dt exists and is analytic on Re z>.assume that G has an analytic extension to a neighborhood of the imaginary axis, Re z =.Then F (t) dt exists as an improper integral and is equal to G().[In fact, F (t)e iyt dt converges for every y R to G(iy).] Results like (7.3.) are named for A. Tauber, who is credited with proving the first theorem of this type near the end of the 9th century.the phrase Tauberian theorem was coined by G.H. Hardy, who along with J.E. Littlewood made a number of contributions in this area.generally, Tauberian theorems are those in which some type of ordinary convergence (e.g., convergence of F (t)e iyt dt for each y R), is deduced from some weaker type of convergence (e.g., convergence of F (t)e zt dt for each z with Re z > ) provided additional conditions are satisfied (e.g., G has an analytic extension to a neighborhood of each point on the imaginary axis).tauber s original theorem can be found in The Elements of Real Analysis by R.G. Bartle Corollary Let f be a nonnegative, piecewise continuous and nondecreasing function on [, ) such that f(x) = O(x).Then its Mellin transform g(z) =z f(x)x z dx exists for Re z> and defines an analytic function g.assume that for some constant c, the function g(z) c z has an analytic extension to a neighborhood of the line Re z =.Then as x, f(x) x c. As stated earlier, we are first going to see how the prime number theorem follows from (7.3.) and (7.3.2). To this end, let ψ be as above, namely ψ(x) = [ ] ln x. p x
11 7.3. PROOF OF THE PRIME NUMBER THEOREM Then ψ is a nonnegative, piecewise continuous, nondecreasing function on [, ).Furthermore, by (7.2.2), ψ(x) =O(x), so by (7.3.2) we may take f = ψ and consider the Mellin transform g(z) =z ψ(x)x z dx. But by (7..4), actually g(z) = ζ (z)/ζ(z), and by the discussion leading up to the ζ statement of (7..4), (z) ζ(z) + z has an analytic extension to a neighborhood of each pointofrez =, hence so does g(z) z. Consequently, by (7.3.2), we can conclude that ψ(x)/x, which, by (7.2.), is equivalent to the PNT. Thus we are left with the proof of (7.3.) and its corollary (7.3.2). Proof of (7.3.) Let F be as in the statement of the theorem. Then it follows ust as in the proof of (7..2), the extension theorem for zeta, that F s Laplace transform G is defined and analytic on Re z >.Assume that G has been extended to an analytic function on a region containing Re z.since F is bounded we may as well assume that F (t),t. For <λ<, define G λ (z) = λ F (t)e zt dt. By (3.3.3), each function G λ is entire, and the conclusion of our theorem may be expressed as lim G λ() = G(). λ That is, the improper integral F (t) dt exists and converges to G().We begin the analysis by using Cauchy s integral formula to get a preliminary estimate of G λ () G(). For each R>, let δ(r) > be so small that G is analytic inside and on the closed path. ir γ  R γ + R δ(r).. δ γ γ + R = R + γ  R. ir Figure 7.3. γ R in Figure (Note that since G is analytic on an open set containing Re z,
12 2 CHAPTER 7. THE PRIME NUMBER THEOREM such a δ(r) > must exist, although it may well be the case that δ(r) asr +.) Let γ + R denote that portion of γ R that lies in Re z>, and γ R the portion that lies in Re z<.by Cauchy s integral formula, G() G λ () = (G(z) G λ (z)) dz. () 2πi γ R z Let us consider the consequences of estimating G() G λ () by applying the usual M L estimates to the integral on the right hand side of () above.first, for z γ + R and x =Rez, we have G(z) G λ (z) z = R F (t)e zt dt R R λ λ λ e λx F (t) e xt dt e xt dt = R x R x = R Re z. But / Re z is unbounded on γ + R, so we see that a more delicate approach is required to shows that G() G λ () asλ.indeed, it is here that Newman s ingenuity comes to the fore, and provides us with a modification of the above integral representation for G() G λ ().This will furnish the appropriate estimate.newman s idea is to replace the factor /z by (/z)+(z/r 2 ) in the path integral in ().Since (G(z) G λ (z))z/r 2 is analytic, the value of the path integral along γ R remains unchanged.we further modify () by replacing G(z) and G λ (z) by their respective products with e λz.since e λz is entire and has the value at z =, we can write G() G λ () = (G(z) G λ (z))e λz ( 2πi γ R z + z R 2 ) dz. Note that for z = R we have (/z)+(z/r 2 )=(z/ z 2 )+(z/r 2 ) = (2 Re z)/r 2, so that if z γ + R, (recalling (2) above), (G(z) G λ (z))e λz ( z + z R 2 ) Re z e λ Re z λ Re z 2Rez e R 2 = 2 R 2. Consequently, (G(z) G λ (z))e λz ( 2πi γ + z + z dz R 2 ) R R by the ML theorem.note that this estimate of the integral along the path γ + R is independent of λ.now let us consider the contribution to the integral along γ R of the integral (2)
13 7.3. PROOF OF THE PRIME NUMBER THEOREM 3 along γ R.First we use the triangle inequality to obtain the estimate (G(z) G λ (z))e λz ( 2πi γ z + z dz R 2 ) R G(z)e λz ( 2πi γ z + z R 2 ) dz + G λ (z)e λz ( 2πi R γ z + z dz R 2 ) R = I (R) + I 2 (R). First consider I 2 (R).Since G λ (z) is an entire function, we can replace the path of integration γ R by the semicircular path from ir to ir in the left half plane.for z on this semicircular arc, the modulus of the integrand in I 2 (R) is ( λ F (t)e zt λz 2Rez dt)e R 2 ) 2 Re z Re z R 2 = 2 R 2. (Note that F, we can replace the upper limit of integration by, and e λx for x.) This inequality also holds if Re z = (let z iy).thus by the ML theorem we get I 2 (R) (/2π)(2/R 2 )(πr) =/R, again. Finally, we consider I (R).This will be the trickiest of all since we only know that on γ R, G is an analytic extension of the explicitly defined G in the right half plane.to deal with this case, first choose a constant M(R) > such that G(z) M(R) for z γ R. Choose δ such that <δ <δ(r) and break up the integral defining I (R) intotwo parts, corresponding to Re z< δ and Re z δ.the first contribution is bounded in modulus by 2π M(R)e λδ ( δ(r) + R )πr = 2 RM(R)( δ(r) + R )e λδ, which for fixed R and δ tends to as λ.on the other hand, the second contribution is bounded in modulus by 2π M(R)( δ(r) + R )2R arcsin δ R, the last factor arising from summing the lengths of two short circular arcs on the path of integration.thus for fixed R and δ(r) we can make the above expression as small as we please by taking δ sufficiently close to.so at last we are ready to establish the conclusion of this theorem.let ɛ> be given.take R =4/ɛ and fix δ(r), <δ(r) <R, such that G is analytic inside and on γ R.Then as we saw above, for all λ, (G(z) G λ (z))e λz ( 2πi γ + z + z dz R 2 ) R = ɛ 4 R and also 2πi γ R (G λ (z)e λz ( z + z R 2 ) dz R = ɛ 4.
14 4 CHAPTER 7. THE PRIME NUMBER THEOREM Now choose δ such that <δ <δ(r) and such that Since 2π M(R)( δ(r) + R )2R arcsin δ R < ɛ 4. 2 RM(R)( δ(r) + R )e λδ < ɛ 4 for all λ sufficiently large, say λ λ, it follows that which completes the proof. Proof of (7.3.2) G λ () G() <ɛ, λ λ Let f(x) and g(z) be as in the statement of the corollary.define F on [, + ) by F (t) =e t f(e t ) c. Then F satisfies the first part of the hypothesis of the auxiliary Tauberian theorem, so let us consider its Laplace transform, G(z) = which via the change of variables x = e t becomes G(z) = = (e t f(e t ) c)e zt dt, ( x f(x) c)x z dx x f(x)x z 2 dx c x z dx = f(x)x z 2 dx c z g(z +) = z + c z = z + [g(z +) c z c]. It follows from the hypothesis that g(z +) (c/z) has an analytic extension to a neighborhood of the line Re z =, and consequently the same is true of the above function G. Thus the hypotheses of the auxiliary Tauberian theorem are satisfied, and we conclude that the improper integral F (t) dt exists and converges to G().In terms of f, this says that (e t f(e t ) c) dt exists, or equivalently (via the change of variables x = e t once more) that ( f(x) x c)dx x
15 7.3. PROOF OF THE PRIME NUMBER THEOREM 5 exists.recalling that f is nondecreasing, we can infer that f(x)/x c as x.for let ɛ> be given, and suppose that for some x >, [f(x )/x ] c 2ɛ.It follows that Hence, f(x) f(x ) x (c +2ɛ) x(c + ɛ) for x x c +2ɛ c + ɛ x. c+2ɛ c+ɛ x x ( f(x) x c)dx x c+2ɛ c+ɛ x x ɛ +2ɛ dx = ɛ ln(c x c + ɛ ). But x 2 x c) dx x asx,x 2, because the integral from to is convergent. Thus for all x sufficiently large, x ( f(x) c+2ɛ c+ɛ x x ( f(x) x c)dx x <ɛln(c +2ɛ c + ɛ ). However, reasoning from the assumption that [f(x )/x ] c 2ɛ, we have ust deduced the opposite inequality.we must conclude that for all x sufficiently large, [f(x )/x ] c <2ɛ. Similarly, [f(x )/x ] c> 2ɛ for all x sufficiently large.[say [f(x )/x ] c 2ɛ. The key inequality now becomes f(x) f(x ) x (c 2ɛ) x(c ɛ) for ( c 2ɛ c ɛ )x x x and the limits of integration in the next step are from c 2ɛ c ɛ x to x.] Therefore f(x)/x c as x, completing the proof of both the corollary and the prime number theorem. The prime number theorem has a long and interesting history.we have mentioned ust a few of the many historical issues related to the PNT in this chapter.there are several other number theoretic functions related to π(x), in addition to the function ψ(x) that was introduced earlier.a nice discussion of some of these issues can be found in Eric W.Weisstein, Prime Number Theorem, from MathWorld A Wolfram Web Resource, This source also includes a number of references on PNT related matters. References.Karl Sabbach, The Riemann HypothesisThe Greatest Unsolved Problem in Mathematics, Farrar, Strass and Giroux, New York, Julian Havil, GammaExploring Euler s Constant, Princeton University Press, Princeton and Oxford, 23, Chapter 5.
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
More information4. Complex integration: Cauchy integral theorem and Cauchy integral formulas. Definite integral of a complexvalued function of a real variable
4. Complex integration: Cauchy integral theorem and Cauchy integral formulas Definite integral of a complexvalued function of a real variable Consider a complex valued function f(t) of a real variable
More informationTHE 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 informationMATH 311: COMPLEX ANALYSIS CONTOUR INTEGRALS LECTURE
MATH 3: COMPLEX ANALYSIS CONTOUR INTEGRALS LECTURE Recall the Residue Theorem: Let be a simple closed loop, traversed counterclockwise. Let f be a function that is analytic on and meromorphic inside. Then
More information1 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 informationThe General Cauchy Theorem
Chapter 3 The General Cauchy Theorem In this chapter, we consider two basic questions. First, for a given open set Ω, we try to determine which closed paths in Ω have the property that f(z) dz = 0for every
More informationDouble Sequences and Double Series
Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine Email: habil@iugaza.edu Abstract This research considers two traditional important questions,
More informationPreliminary Version: December 1998
On the Number of Prime Numbers less than a Given Quantity. (Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse.) Bernhard Riemann [Monatsberichte der Berliner Akademie, November 859.] Translated
More informationReference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.
5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2
More informationUndergraduate 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 informationSeries. 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 informationPower Series. Chapter Introduction
Chapter 10 Power Series In discussing power series it is good to recall a nursery rhyme: There was a little girl Who had a little curl Right in the middle of her forehead When she was good She was very,
More informationU.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 informationk=1 k2, and therefore f(m + 1) = f(m) + (m + 1) 2 =
Math 104: Introduction to Analysis SOLUTIONS Alexander Givental HOMEWORK 1 1.1. Prove that 1 2 +2 2 + +n 2 = 1 n(n+1)(2n+1) for all n N. 6 Put f(n) = n(n + 1)(2n + 1)/6. Then f(1) = 1, i.e the theorem
More informationANALYTIC 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 informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationCourse 221: Analysis Academic year , First Semester
Course 221: Analysis Academic year 200708, 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 informationMOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu
Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing
More information5.3 Improper Integrals Involving Rational and Exponential Functions
Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a
More informationThe Alternating Series Test
The Alternating Series Test So far we have considered mostly series all of whose terms are positive. If the signs of the terms alternate, then testing convergence is a much simpler matter. On the other
More information61. 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 information3.1. Sequences and Their Limits Definition (3.1.1). A sequence of real numbers (or a sequence in R) is a function from N into R.
CHAPTER 3 Sequences and Series 3.. Sequences and Their Limits Definition (3..). A sequence of real numbers (or a sequence in R) is a function from N into R. Notation. () The values of X : N! R are denoted
More informationSequences 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 informationANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series
ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a 3 + + + where { } is a sequence of real numbers. Here formal means that we don
More informationThe positive integers other than 1 may be divided into two classes, prime numbers (such as 2, 3, 5, 7) which do not admit of resolution into smaller
The positive integers other than may be divided into two classes, prime numbers (such as, 3, 5, 7) which do not admit of resolution into smaller factors, and composite numbers (such as 4, 6, 8, 9) which
More informationThe Goldberg Rao Algorithm for the Maximum Flow Problem
The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }
More informationLimit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)
SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f
More information5. 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,
More informationContinued 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 information6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.
hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,
More information6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )
6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a nonempty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points
More informationMathematical 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 informationChapter 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
More informationStructure and randomness in the prime numbers. Science colloquium January 17, A small selection of results in number theory. Terence Tao (UCLA)
Structure and randomness in the prime numbers A small selection of results in number theory Science colloquium January 17, 2007 Terence Tao (UCLA) 1 Prime numbers A prime number is a natural number larger
More informationLecture 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 informationFourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks
Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results
More informationPrime numbers and prime polynomials. Paul Pollack Dartmouth College
Prime numbers and prime polynomials Paul Pollack Dartmouth College May 1, 2008 Analogies everywhere! Analogies in elementary number theory (continued fractions, quadratic reciprocity, Fermat s last theorem)
More information2 Complex Functions and the CauchyRiemann Equations
2 Complex Functions and the CauchyRiemann Equations 2.1 Complex functions In onevariable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)
More informationAgain, the limit must be the same whichever direction we approach from; but now there is an infinity of possible directions.
Chapter 4 Complex Analysis 4.1 Complex Differentiation Recall the definition of differentiation for a real function f(x): f f(x + δx) f(x) (x) = lim. δx 0 δx In this definition, it is important that the
More informationTHE 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......................................
More informationSummer High School 2009 Aaron Bertram. 1. Prime Numbers. Let s start with some notation:
Summer High School 2009 Aaron Bertram 1. Prime Numbers. Let s start with some notation: N = {1, 2, 3, 4, 5,...} is the (infinite) set of natural numbers. Z = {..., 3, 2, 1, 0, 1, 2, 3,...} is the set of
More informationChapter 2 Sequences and Series
Chapter 2 Sequences and Series 2. Sequences A sequence is a function from the positive integers (possibly including 0) to the reals. A typical example is a n = /n defined for all integers n. The notation
More informationLecture 3. Mathematical Induction
Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion
More informationPERIMETERS OF PRIMITIVE PYTHAGOREAN TRIANGLES. 1. Introduction
PERIMETERS OF PRIMITIVE PYTHAGOREAN TRIANGLES LINDSEY WITCOSKY ADVISOR: DR. RUSS GORDON Abstract. This paper examines two methods of determining whether a positive integer can correspond to the semiperimeter
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationSection 3 Sequences and Limits
Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the nth term of the sequence.
More informationPRIME 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 informationCONTRIBUTIONS 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
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationPrime 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 informationMATH : HONORS CALCULUS3 HOMEWORK 6: SOLUTIONS
MATH 1630033: HONORS CALCULUS3 HOMEWORK 6: SOLUTIONS 251 Find the absolute value and argument(s) of each of the following. (ii) (3 + 4i) 1 (iv) 7 3 + 4i (ii) Put z = 3 + 4i. From z 1 z = 1, we have
More informationGeneral Theory of Differential Equations Sections 2.8, 3.13.2, 4.1
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.13.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions
More informationTaylor 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
More informationMATH 361: NUMBER THEORY FIRST LECTURE
MATH 361: NUMBER THEORY FIRST LECTURE 1. Introduction As a provisional definition, view number theory as the study of the properties of the positive integers, Z + = {1, 2, 3, }. Of particular interest,
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More information1.3 Induction and Other Proof Techniques
4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.
More informationMATH 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 informationRAMANUJAN S HARMONIC NUMBER EXPANSION INTO NEGATIVE POWERS OF A TRIANGULAR NUMBER
Volume 9 (008), Issue 3, Article 89, pp. RAMANUJAN S HARMONIC NUMBER EXPANSION INTO NEGATIVE POWERS OF A TRIANGULAR NUMBER MARK B. VILLARINO DEPTO. DE MATEMÁTICA, UNIVERSIDAD DE COSTA RICA, 060 SAN JOSÉ,
More informationSums of Independent Random Variables
Chapter 7 Sums of Independent Random Variables 7.1 Sums of Discrete Random Variables In this chapter we turn to the important question of determining the distribution of a sum of independent random variables
More information1 Review of complex numbers
1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely
More informationThe 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 informationx 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 informationOn the Sign of the Difference ir( x) li( x)
mathematics of computation volume 48. number 177 january 1987. pages 323328 On the Sign of the Difference ir( x) li( x) By Herman J. J. te Riele Dedicated to Daniel Shanks on the occasion of his 10 th
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationCourse 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
More information1. Periodic Fourier series. The Fourier expansion of a 2πperiodic function f is:
CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2πperiodic 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
More informationFunction Sequences and Series
Chapter 7 Function Sequences and Series 7.1 Complex Numbers This is not meant to be a thourogh and completely rigorous construction of complex numbers but rather a quick brush up of things assumed to be
More informationInfinite series, improper integrals, and Taylor series
Chapter Infinite series, improper integrals, and Taylor series. Introduction This chapter has several important and challenging goals. The first of these is to understand how concepts that were discussed
More information3 Contour integrals and Cauchy s Theorem
3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of
More information3. 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.
More informationContinued fractions and good approximations.
Continued fractions and good approximations We will study how to find good approximations for important real life constants A good approximation must be both accurate and easy to use For instance, our
More informationSECTION 102 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
More informationThe 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)
More informationAppendix F: Mathematical Induction
Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another
More information7. Cauchy s integral theorem and Cauchy s integral formula
7. Cauchy s integral theorem and Cauchy s integral formula 7.. Independence of the path of integration Theorem 6.3. can be rewritten in the following form: Theorem 7. : Let D be a domain in C and suppose
More informationReducibility of Second Order Differential Operators with Rational Coefficients
Reducibility of Second Order Differential Operators with Rational Coefficients Joseph Geisbauer University of ArkansasFort Smith Advisor: Dr. Jill Guerra May 10, 2007 1. INTRODUCTION In this paper we
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationSUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by
SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples
More informationn k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...
6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence
More informationPrime 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.
More informationMath 317 HW #7 Solutions
Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence
More information1 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
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationFIRST 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
More informationMATH31011/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
More informationLecture 13  Basic Number Theory.
Lecture 13  Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are nonnegative integers. We say that A divides B, denoted
More informationStirling s formula, nspheres and the Gamma Function
Stirling s formula, nspheres and the Gamma Function We start by noticing that and hence x n e x dx lim a 1 ( 1 n n a n n! e ax dx lim a 1 ( 1 n n a n a 1 x n e x dx (1 Let us make a remark in passing.
More informationProblem 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 informationRAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS PART A
RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:
More informationTHE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION
THE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION ERICA CHAN DECEMBER 2, 2006 Abstract. The function sin is very important in mathematics and has many applications. In addition to its series epansion, it
More informationCounting 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
More informationPART I. THE REAL NUMBERS
PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS
More informationAPPLICATIONS OF THE ORDER FUNCTION
APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and
More informationApplications of Fourier series
Chapter Applications of Fourier series One of the applications of Fourier series is the evaluation of certain infinite sums. For example, n= n,, are computed in Chapter (see for example, Remark.4.). n=
More informationChapter 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 information5 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
More informationTaylor 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
More informationChapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1
Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes
More informationSome 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
More information