SOME IDENTITIES FOR THE RIEMANN ZETA-FUNCTION II. Aleksandar Ivić

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1 FACTA UNIVERSITATIS (NIŠ Ser. Math. Inform. 2 (25, 8 SOME IDENTITIES FOR THE RIEMANN ZETA-FUNCTION II Aleksandar Ivić Abstract. Several identities for the Riemann zeta-function ζ(s are proved. For eample, if ϕ ( := {} = [], ϕ n ( := ( {u}ϕ u n u (n 2, then ζ n (s + ( s n = ϕ n ( s d (s = σ + it, < σ < and + ζ(σ + it 2n 2π (σ 2 + t 2 n dt = ϕ 2 n( 2σ d ( < σ <. Let as usual ζ(s = + n= n s (Re s > denote the Riemann zetafunction. This note is the continuation of the author s work [6], where several identities involving ζ(s were obtained. The basic idea is to use properties of the Mellin transform (f : [, + R ( F (s = M[f(; s] := f( s d (s = σ + it, σ >, in particular the analogue of the Parseval formula for Mellin transforms, namely (2 2πi σ+i σ i F (s 2 ds = f 2 ( 2σ d. For the conditions under which (2 holds, see e.g., [5] and []. If {} denotes the fractional part of ({} = [], where [] is the greatest integer not Received June 7, Mathematics Subject Classification. Primary M6.

2 2 A. Ivić eceeding, we have the classical formula (see e.g., eq. Titchmarsh [2] (2..5 of E.C. (3 ζ(s s = {} s d = {/} s d, where s = σ + it, < σ <. A quick proof is as follows. We have ζ(s = = s = s s d[] = s ([] s d + s {} s d + [] s d s s. s d This holds initially for σ >, but since the last integral is absolutely convergent for σ >, it holds in this region as well by analytic continuation. Since s {} s d = s s d = s ( < σ <, s we obtain (3 on combining the preceding two formulae. We note that (3 is a special case of the so-called Müntz s formula (with f( = χ [,] (, the characteristic function of the unit interval (4 ζ(sf (s = P f( s d, where the Müntz operator P is the linear operator defined formally on functions f : [, + C by (5 P f( := n= f(n f(t dt. Besides the original proof of (4 by Müntz [8], proofs are given by E.C. Titchmarsh [2, Chapter, Section 2.] and recently by L. Báez-Duarte [2]. The identity (4 is valid for < σ < if f ( is continuous, bounded in any finite interval and is O( β for where β > is a constant. The identity (3, which Báez-Duarte [2] calls the proto-müntz identity, plays an important rôle in the approach to the Riemann Hypothesis (RH, that

3 Some Identities for the Riemann Zeta-function II 3 all comple zeros of ζ(s have real parts equal to /2 via methods from functional analysis (see e.g., the works [] [4] and [9]. Our first aim is to generalize (3. We introduce the convolution functions ϕ n ( by (6 ϕ ( := {} = [], ϕ n ( := ( {u}ϕ n u u The asymptotic behaviour of the function ϕ n ( is contained in Theorem. If n 2 is a fied integer, then ( (7 ϕ n ( = (n! logn (/ + O log n 2 (/ and (8 ϕ n ( = O(log n ( + (. (n 2. ( < <, Proof. Using the properties of {}, namely {} = for < < and {}, one easily verifies (7 and (8 when n = 2. To prove the general case we use induction, supposing that the theorem is true for some n. Then, when < <, ϕ n+ ( = + + = I + I 2 + I 3, say. We have, by change of variable, I = {u}ϕ n ( u u = By the induction hypothesis I 2 = = = (n! ( ϕ n du = u ϕ n (v dv v 2 ( ( {u}ϕ n u u = ϕ n du u { ( u ( ( u } (n!u logn + O u logn 2 du / log n y dy ( y + O log n (/ = n! logn (/ + O( log n (/. = O(.

4 4 A. Ivić Finally, since {} and (8 holds, I 3 = {u}ϕ n ( u u The proof of (8 is on similar lines, when we write ϕ n+ ( = + ( log n u u 2 logn + = J + J 2 + J 3 (, (. say, so that there is no need to repeat the details. By more elaborate analysis (7 could be further sharpened. Theorem 2. If n is a fied integer, and s = σ + it, < σ <, then (9 ζ n (s ( s n = ϕ n ( s d. Clearly (9 reduces to (3 when n =. From Theorem it transpires that the integral in (9 is absolutely convergent for < σ <. By using (2 (with s in place of s we obtain the following Corollary. For n N we have ( + ζ(σ + it 2n 2π (σ 2 + t 2 n dt = ϕ 2 n( 2σ d ( < σ <. Proof of Theorem 2. As already stated, (9 is true for n =. The general case is proved then by induction. Suppose that (9 is true for some n, and consider ζ n+ (s ( s n+ = {}ϕ n (y(y s d dy ( < σ < as a double integral. We make the change of variables = v, y = u/v, noting that the absolute value of the Jacobian of the transformation is /v. The above integral becomes then {v}ϕ n ( u v u s v du dv = = ( {v}ϕ n ( u v ϕ n+ ( s d, dv u s du v

5 Some Identities for the Riemann Zeta-function II 5 as asserted. The change of integration is valid by absolute convergence, which is guaranteed by Theorem. Remark. L. Báez-Duarte kindly pointed out to me that the above procedure leads in fact formally to a convolution theorem for Mellin transforms, namely (cf. ( [ ( M f(ug ( ] u u ; s = M[f(; s] M[g(; s] = F (sg(s, which is eq. ( of I. Sneddon []. An alternative proof of (9 follows from the second formula in (3 and (, but we need again a result like Theorem to ensure the validity of the repeated use of (. A similar approach via (modified Mellin transforms and convolutions was carried out by the author in [7]. There is another possibility for the use of the identity (3. Namely, one can evaluate the Laplace transform of {}/ for real values of the variable. This is given by Theorem 3. If M is a fied integer and γ denotes Euler s constant, then for T + (2 {} e /T d = 2 log T 2 γ + 2 log(2π + M m= ζ( 2m (2m!( 2m T 2m + O M (T 2M. Proof. We multiply (3 by T s Γ(s, where Γ(s is the gamma-function, integrate over s and use the well-known identity (e.g., see the Appendi of [5] e z = c+i z s Γ(s ds (Re z >, c >. 2πi We obtain c i (3 c+i ζ(s + 2πi c i s T s {} Γ(s ds = e /T d ( < c <. In the integral on the left-hand side of (3 we shift the line of integration to Re s = N /2, N = 2M + (i.e., taking c = N /2 and then apply the residue theorem. The gamma-function has simple poles at s = m, m =,, 2,... with residues ( m /m!. The zeta-function has

6 6 A. Ivić simple (so-called trivial zeros at s = 2m, m N, which cancel with the corresponding poles of Γ(s. Thus there remains a pole of order two at s =, plus simple poles at s =, 3, 5,.... The former produces the main term in (2, when we take into account that ζ( = 2, ζ ( = 2 log(2π (see [5, Chapter ] and Γ ( = γ. The simple poles at s =, 3, 5,... produce the sum over m in (2, and the proof is complete. Remark 2. The method of proof clearly yields also, as T +, ϕ n ( e /T d = P n (log T + M c m,n T 2m + O M (T 2M, m= where P n (z is a polynomial in z of degree n whose coefficients may be eplicitly evaluated, and c m,n are suitable constants which also may be eplicitly evaluated. For our last result we turn to Müntz s identity (4 (5 and choose f( = e π2, which is a fast converging kernel function. Then P f( = F (s = n= f(n f(t dt = n= e π2 s d = 2 π s/2 Γ( 2 s. From (2 and (4 it follows then that, for < σ <, e πn2 2 2, (4 ζ(σ + itγ( 2 σ + 2 it 2 dt ( + + = 8π +σ n= 2 e πn2 2 2σ d. 2 The series on the right-hand side of (4 is connected to Jacobi s theta function (5 θ(z := e πn2 z n= (Re z >, which satisfies the functional equation (proved easily by e.g., Poisson summation formula (6 θ(t = ( θ (t >. t t

7 Some Identities for the Riemann Zeta-function II 7 From (5 (6 we infer that (7 n= e πn2 2 = 2 ( θ( 2 = ( 2 θ 2 2 ( >. By using (7 it is seen that the right-hand side of (4 equals (8 8π +σ ( ( 4 θ 2 2 2σ d = 2π +σ (uθ(u 2 u 2 u 2σ du. The (absolute convergence of the last integral at infinity follows from uθ(u 2 u = 2u n= while the convergence at zero follows from e πn2 u 2, ( ( uθ(u 2 = θ u 2 = + O e u 2 (u +. Now we note that (4 remains unchanged when σ is replaced by σ, and then we use the functional equation (see e.g., [5, Chapter ] for ζ(s in the form π s/2 ζ(sγ( 2 s = π ( s/2 ζ( sγ( 2 ( s to transform the resulting left-hand side of (4. Then (4 and (8 yield the following Theorem 4. For < σ < we have ζ(σ + itγ( 2 σ + 2 it 2 dt = 2π σ (uθ(u 2 u 2 u 2σ 3 du.

8 8 A. Ivić R E F E R E N C E S. L. Báez-Duarte: A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9 Mat. Appl. 4 (23, no., L. Báez-Duarte: A general strong Nyman-Beurling criterion for the Riemann Hypothesis. Publ. Inst. Math. (Beograd (N.S. (to appear. 3. L. Báez-Duarte, M. Balazard, B. Landreau et E. Saias: Notes sur la fonction ζ de Riemann, 3. Adv. Math. 49 (2, A. Beurling: A closure problem related to the Riemann zeta-function. Proc. Nat. Acad. Sci. U.S.A. 4 (955, A. Ivić: The Riemann Zeta-function. John Wiley & Sons, New York, A. Ivić: Some identities for the Riemann zeta-function. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 4 (23, A. Ivić: Estimates of convolutions of certain number-theoretic error terms. Int. J. Math. Math. Sci. 24, no. -4, C. H. Müntz: Beziehungen der Riemannschen ζ-funktion zu willkürlichen reellen Funktionen. Mat. Tidsskrift, B (922, B. Nyman: On the One-Dimensional Translation Group and Semi- Group in Certain Function Spaces. Thesis, University of Uppsala, 95, 55 pp.. I. Sneddon: The Use of Integral Transforms. McGraw-Hill, New York etc., E. C. Titchmarsh: Introduction to the Theory of Fourier Integrals. Oford University Press, Oford, E. C. Titchmarsh: The Theory of the Riemann Zeta-Function. (2nd ed., Oford at the Clarendon Press, 986. Katedra Matematike RGF-a Universiteta u Beogradu D ušina 7, Beograd Serbia and Montenegro e mail: aivic@matf.bg.ac.yu, ivic@rgf.bg.ac.yu