Riemann s ζ-function

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 4, 9, no. 9, Riemann s ζ-function R. A. Mollin Department of Mathematics and Statistics University of Calgary, Calgary, Alberta, Canada, TN N4 URL: ramollin/ ramollin@math.ucalgary.ca Abstract We provide an overview of the Riemann ζ-function with an eye to a proof of the functional equation that is non-standard and more accessible than those proofs readily available in modern texts. This is s survey note with the goal of a more elegant and transparent presentation of the salient features surrounding the functional equation for the ζ-function and the intimate connections with the Riemann hypothesis. Strikingly, Riemann published exactly one paper [3] on number theory, appearing in 859. This landmark paper changed the perspective of the subject immediately. His approach was elegant and profound in its approach to the distribution of prime numbers. He basically took Cauchy s theory of holomorphic functions and appealed to the process, discovered by Cauchy, of looking at each point of an analytic function as an average of its values at neighbouring points. This was calculated as a path integral along a closed curve in C, which permitted deformations of the path of integration allowing an approximation of the integral. Riemann took Euler s formula, described below, for primes and extended it to C. This was accomplished via the Riemann ζ-function given for s = a+ bi C with Res =a>by ζs = n= n s = p s. The last equality is known as the Euler product, which provides a fundamental relationship between the primes and the zeta function. The series on the left is absolutely convergent, which implies that ζs is analytic the half plane Res >. To show that ζs may be analytically continued to the

2 46 R. A. Mollin entire complex plane with one singularity this we may employ the following well-known result from any standard text in analysis. The symbol B j denotes the jth Bernoulli number in what follows, where the Taylor series for x C is given by x e x = B j x j. j! Also, f j x will denote the jth derivative of a given function fx. Some of the following is adapted from [] and all relevant background material may be found in most texts on introductory number theory such as [], for instance. Theorem The Euler-Maclaurin Sum Formula Let a<bare integers and n N. If fx has a continuous derivative for x [a, b], then j= b fj = b j=a+ a fxdx + n i= i B i i! f i b f i a + n n! b a B n x x f n xdx. Theorem For s C and Res > n, for n N, ζs is convergent, except at s =, and ζs = s + + n j= B j ss + s + j j! n! ss + s + n B n t t t s n dt. Proof. Let n N and set fx =x s, a =, and b = N in Theorem. Since f n x = n ss + s + s + n x s n and Therefore, ζs = + lim N N fj. j= [ N ζs = lim N x s dx N j= B j j! ss + s + j N s+ j

3 Riemann s ζ-function 47 = lim N ] N n! ss + s + n B n x x x s n dt [ N s + N s s n B j j! ss + s + j N s j+ j= ] N n! ss + s + n B n x x x s n dx. For Res >, we may pull the limit through, so since lim N s = = lim N s, N N we get, the right-hand side of which converges for Res > n, except at s =. Now we turn to the relationship between the Riemann ζ-function and the distribution of primes. We begin by reminding the reader that πx denotes the number of primes less than x. The first celebrated result is our starting point. Theorem 3 The Prime Number Theorem PNT For x R +, πx x log e x. The close relationship between the Riemann ζ-function and πx is given by πx log e ζs =s dx, 3 xx s for Res >. It is noteworthy that the Euler product for the Riemann ζ-function tells us that since ζs as s then there are infinitely many primes. To see this consider ζs for s R +. By the series expansion in, ζs diverges as s + since the harmonic series /j j= diverges. Actually, more can be said. Since log e p s = log e p s

4 48 R. A. Mollin = p s + O < p + O when s>, then as s +, the sum p diverges. Understanding the sums p x p is implicit in the development of Theorem 3. Indeed, the following predates the PNT, and follows from it. Theorem 4 Merten s Theorem p x where γ is Euler s constant and M = γ + called Merten s constant. p = log e log e x + M + o, p x log e +, p p Note that Theorem 4 is equivalent to the asymptotic relationship e γ p log e x. There is an equivalent formulation of Theorem 3 via the following function called Merten s function. Mx = n x μx, for any x R, where μ is the Möbius function. It can be shown that Theorem 3 is equivalent to the following. Theorem 5 Merten s Equivalence to the PNT Mx =ox. Even more, Theorem 3 is also equivalent to the following. Theorem 6 Möbius Equivalence to the PNT n= μn n =.

5 Riemann s ζ-function 49 The relationship between the Riemann ζ-function and Merten s function is given by ζs = μd Mx = s dx, d s d= xs+ for Res >. This brings us to one of the most important and celebrated outstanding problems. Conjecture The Riemann Hypothesis RH All of the zeros of ζs in the critical strip <Res < lie on the critical line Res =/. In terms of the Merten s function we may reformulate Conjecture as being equivalent to the following. Conjecture Merten s Equivalence to the RH Mx =O ε x +ε, for any fixed ε>, where O ε means that, in the big O notation, the constant depends on ε only. Also, Riemann conjecture the following in 859, which is also equivalent to Conjecture. Conjecture 3 Integral Equivalence to the RH where called the logarithmic integral. πx = lix+o x log e x, lix = dt log e t, Conjecture 3 will hold if and only if the Riemann ζ-function does not vanish on the half plane Res > /. In other words, Conjecture is equivalent to the statement that the error which occurs, when πx is estimated by lix, is O x log e x. Now we are in a position to establish a fundamental equation, which puts the above more in focus. Indeed, the following shows the central role that the Riemann ζ-function plays in analytic number theory via the functional equation, ζs =fsζ s, where we we define fs below.

6 4 R. A. Mollin Let n = 3 in to get, ζs = s + + B s ss + s + B 3 t t t s 3 dt, 6 for Res >. An exercise involving integration by parts three times and the Bernoulli relationship yields so ss + s + 6 B n+ x =n +B nx, B 3 t t t s 3 dt = B s s, 4 ss + s + ζs = B 3 t t t s 3 dt. 6 Replacing B 3 t t by the Fourier series, B k x x = k k! we get ss + s + ζs = 6 and by setting x =πjt, this equals, ss + s + j= j= πj s j= sinπjx, for k, πj k sinπjt t s 3 dt πj 3 x s 3 sin x dx. Since sin x = i x i+ i +! i= is the analytic continuation of the usual trigonometric function, and converges for all x C, then we may interchange the sum and integral above so the latter equals s π s ss + s + x s 3 sin xdx j= j s. Hence, ζs = s π s ss + s + x s 3 sin xdx ζ s. 5 In order to complete the derivation of the functional equation, we need the following concept due to Euler.

7 Riemann s ζ-function 4 Definition The Gamma Function For s C and Res > the gamma function is given by Γs = e t t s dt. We will employ two well-known formulas for the gamma function given as follows. For <Rez <, the Wolfskehl equation is given by sin πz Γ z =z a formula known since 886 see [5], and Now we are ready for the functional equation. y z sin ydy, 6 zγ z = Γ z. 7 Theorem 7 Riemann s Functional Equation for ζs For s C, ζs = s π s Γ sζ s. Proof. From 5, we have to show only that ss + s + x s 3 sin xdx= Γ s. To this end, we employ 6 7 as follows, ss + s + = ss + = s x s 3 sin xdx= ss + Γ s = s Γ s = and we have our functional equation. sγ s = Γ s + s + Γ s + Γ s,

8 4 R. A. Mollin Remark The functional equation is valid for all complex numbers s where both sides are defined. We know that ζs has no zeros for Rs and has only trivial zeros for Rs, which correspond to poles of Γs/, and has infinitely many zeros on the critical strip < Rs <. We may define a related function, which shows symmetry properties more readily than does Theorem 7. If we define ξs =π s/ Γs/ζs, then it may be shown that s ξs =π s/ Γ ζ s =ξ s 8 showing that ξs is symmetric about the critical line Rs =/. Note that ξs is analytic on the whole plane such functions are called entire, since the factor of s eliminates the pole of ζs at s =. Often ξs is called the completed zeta function. The functional equation given in Theorem 7 shows that if s is a zero in the critical strip, then so is s, since by Theorem, zeros occur in complex conjugate pairs. So if it were to be incredibly that the Reimann hypothesis is false, then zeros in the critical strip that are not on the critical line would occur in four-tuples corresponding to vertices of rectangles in the complex plane. The zeros of the ζ-function are intimately connected with the distribution of primes. If U denotes the upper bound of the real parts of the zeros of ζs, with / U, then πx lix cx U log e x for a constant c R +. The Riemann hypothesis is tantamount to U =/. Furthermore, as discussed on page?? in reference to transcendence, the values of ζn+ are largely a mystery, with the exception of a notable Apéry s constant ζ3 = 5 j j 3 j Q. j j= Remark Another rather interesting application of the ζ-function is the following. It can shown that that the probability of two randomly selected integers being relatively prime is approximately equal to /ζ = 6/π = This is also the probability that a randomly selected integer is squarefree. The reason the latter is true, in terms of the Riemann ζ-function given by the Euler product in, is that for a number to be squarefree it must not be divisible by the same prime more than once. In other words, either it is not divisible by p or it is divisible by p but not divisible by it again. Thus, the probability that an integer is not divisible by the square of a prime p equals p + p p = p,

9 Riemann s ζ-function 43 and taking the product over all primes assuming the independence of the divisibility by different primes the probability then that an integer is squarefree tends to p = ζ. This has a generalization, which can be shown by the same reasoning as for n =, namely that the probability that n randomly selected integers are coprime is P n ζn 9 Thus, we may calculate P 3.83, P 4 9/π.939, and so forth. Again using similar reasoning to the above, the probability that a randomly selected integer is cube-free, or fourth-power free, etc., is also given by 9. Thus, the probability that a randomly selected integer is cube-free equals roughly 83%, and that an integer is fourth-power free is roughly 9%. A more general question still, and one that is not resolved at the time of this writing is the following. What is the probability that n randomly selected integers have greatest common divisor equal to g? Let this probability be denoted by P n g see [4, page 48]. Conjecture 4 P n g g n ζn. We have demonstrated the validity of Conjecture 4 for g =, which is 9. The problem remains open. The above is a modest journey through the great depth of the contributions by Riemann to analytic number theory, but provides an overview of some rather pleasant connections from a slightly different and perhaps more accessible perspective. Acknowledgements: The author s research is supported by NSERC Canada grant # A8484. References [] R.A. Mollin, Fundamental Number Theory with Applications, second edition, Chapman and Hall/CRC, Taylor and Francis Group, Boca Raton, London, New York 8.

10 44 R. A. Mollin [] R.A. Mollin, Advanced Number Theory, Chapman and Hall/CRC, Taylor and Francis Group, to appear in 9. [3] B. Riemann, Uber die Anzahl der Primzahlen unter einer gegbren Grösse, Monatsberichte der Berliner Akadamie 859. [4] M.R. Schroeder, Number Theory in Science and Communication, Springer 999. [5] P. Wolfskehl, Beweis, dass der zweite Factor der Klassenzahl für die aus den elfen und dreizehnten Einheitswurzeln gebildeten Zahlen gleich Eins ist, J. Reine Angew Math., Received: December, 8