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Journal of Computational and Applied Mathematics 233 (2009) 667–673 View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 233 (2009) 667–673 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Summation properties of the ηj and Li constants Mark W. Coffey Department of Physics, Colorado School of Mines, Golden, CO 80401, United States article info a b s t r a c t Article history: We find new summatory and other properties of the constants ηj entering the Laurent Received 7 October 2007 expansion of the logarithmic derivative of the Riemann zeta function about s D 1. We relate these constants to other coefficients and functions appearing in the theory of the Keywords: zeta function. In particular, connections to the Li equivalence of the Riemann hypothesis are Binomial transform discussed and quantitatively developed. The validity of the Riemann hypothesis is reduced Li constants to the condition of the sublinear order of a certain alternating binomial sum. Riemann zeta function ' 2009 Elsevier B.V. All rights reserved. Riemann xi function Logarithmic derivatives Riemann hypothesis Li criterion Laurent expansion 1. Introduction f g1 The Riemann hypothesis is equivalent to the Li criterion governing the sequence of real constants λk kD1, that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. This equivalence results from a necessary and sufficient condition that the logarithmic derivative of the function ξT1=.1 − z/U be analytic in the unit disk, where ξ is the Riemann xi function. The Li equivalence [1] states that a necessary and sufficient condition for the nontrivial zeros of D f g1 the Riemann zeta function to lie on the critical line Re s 1=2 is that λk kD1 is nonnegative for every integer k. In this paper we determine summation and other properties of the constants ηj entering the Laurent expansion of the logarithmic derivative of the Riemann zeta function about s D 1. Such properties are key to our approach of lower bounding the Li constants [2,3], since the ηj's are prominent in a subcomponent sum S2 of the Li constants. In fact, should S2.n/ be provably O.n/ or smaller, the Riemann hypothesis would follow. P −j The Li constants are the binomial transform of the sequence fσkg, where σj D ρ ρ are sums of reciprocal powers of the nontrivial zeros of the Riemann zeta function. Herein we recall the relation of the ηj coefficients to the σk's, and use it to provide an alternative proof of a summatory relation among the former constants. It is possible to consider the Li constants as a particular value of differentiable functions defined on the positive real axis. Indeed, Freitas [4] has very recently taken this approach and related the nonnegativity of such functions αn to zero-free half-plane regions of the zeta function. In this paper, we present an explicit representation of the αn.s/'s in terms of the ηj coefficients. Furthermore, we estimate the resulting polygamma summation term S1.n; s/. All of these developments show the context of the s D 1 special case for the Li constants. Since the αj functions satisfy an infinite set of linear first-order ordinary differential equations, it is also possible to describe their derivatives at s D 1 directly in terms of the Li constants. These and related results are also described in a later section of the paper. After introducing some notation and some definitions, we proceed to the summation results for the ηj constants. Here we make use of the functional equation for the zeta function [5–9] and hence of its logarithmic derivative. Our work makes multiple use of the properties of the polygamma function, including the connection of its reflection formula to derivatives of E-mail address: [email protected]. 0377-0427/$ – see front matter ' 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2009.02.034 668 M.W. Coffey / Journal of Computational and Applied Mathematics 233 (2009) 667–673 the cotangent function. From such relations we are able to explicitly construct the necessary expansions with overlapping domains of validity. In this paper, attention is restricted to the classical zeta function. However, the developments carry over to Dirichlet L functions, Hecke L functions, and other generalized zeta functions. The Li constants are defined for such functions [10] and an equivalence relates them to a generalized Riemann hypothesis. For some explicit results and estimations along these lines, Refs. [11,12] may be consulted. 2. Notation and background The function ξ is determined from ζ by the relation ξ.s/ D .s=2/.s−1)π −s=20.s=2)ζ .s/, where 0 is the Gamma function D − f g1 [7,6,9,5,8], and satisfies the functional equation ξ.s/ ξ.1 s/. The sequence λn nD1 is defined by n 1 d n−1 λn D Ts ln ξ.s/UsD1: (1) .n − 1/W dsn The λj's are connected to sums over the nontrivial zeros of ζ .s/ by way of [13,1] n X 1 λn D 1 − 1 − : (2) ρ ρ In the representation [14,15,11] n λn D S1.n/ C S2.n/ − (γ C ln π C 2 ln 2/ C 1; (3) 2 where γ is the Euler constant, the sum n X m n −m S1.n/ ≡ .−1/ .1 − 2 )ζ .m/; n ≥ 2; (4) m mD2 has been bounded [11]: n n 1 n n 1 ln n C (γ − 1/ C ≤ S1.n/ ≤ ln n C (γ C 1/ − : (5) 2 2 2 2 2 2 Further bounds on S1.n/ have been developed by applying Euler–MacClaurin summation to all orders [11]. The summand of the quantity S1 can be written as a sum over the trivial zeros of ζ : 1 −2 − X 1 X 1 .1 − 2 n)ζ .n/ − 1 D D : (6) .2m C 1/n .1 − j/n mD1 j=−∞ even The constants ηj enter the expansion around s D 1 of the logarithmic derivative of the zeta function, 0 1 ζ .s/ 1 X p D − − ηp.s − 1/ ; js − 1j < 3; (7) s s − 1 ζ . / pD0 the corresponding Dirichlet series valid for Re s > 1 is 0 1 ζ .s/ X Λ.n/ D − ; (8) s ns ζ . / nD1 and these constants are used to form the sum S2: Xn n S2.n/ ≡ − ηm−1: (9) m mD1 The constants ηj can be written as [14,16] ! − k N kC1 . 1/ X 1 k ln N ηk D lim Λ.m/ ln m − : (10) kW N!1 m k C 1 mD1 In Eqs. (8) and (10), Λ is the von Mangoldt function [7,6,9,5,8,16], such that Λ.k/ D ln p when k is a power of a prime p and Λ.k/ D 0 otherwise. M.W. Coffey / Journal of Computational and Applied Mathematics 233 (2009) 667–673 669 3. Summation relation for the ηj 's As part of our program to characterize the ηj constants, we derive an explicit summatory relation. We first demonstrate Proposition 1. We have 1 X j 1 − .−1/ ηj D ln 2π; (11) jD0 and for j ≥ 1 1 j j jC1 −.jC1/ X kC1 k T1 C .−1/ Uηj D .−1/ C 1 − T1 C .−1/ U2 ζ .j C 1/ C .−1/ ηk: (12) j kDjC1 Eq. (11) follows simply by evaluating Eq. (7) at s D 0 and using the well known result ζ 0.0)/ζ .0/ D ln.2π/. We prove relation (12) in two different ways. The first method is directly based upon the functional equation satisfied by the Riemann zeta function. The second method relies upon the functional equation of the Riemann xi function. The two paths show the consistency of the theory, and several of the intermediate results may be of independent interest themselves. First approach. From the functional equation satisfied by the classical zeta function, we obtain that of its logarithmic derivative: ζ 0.s/ π π ζ 0.1 − s/ D − .1 − s/ C ln 2π C cot s − : (13) ζ .s/ 2 2 ζ .1 − s/ In this equation, D 00=0 is the digamma function. We develop each term of Eq. (13) in a power series in s − 1. There is a common region of validity of these expansions about s D 1, so that the conclusion (12) holds. From the defining relation (7), use of the binomial theorem, and reordering of series, we have 0 − 1 1 ζ .1 s/ 1 X X j j n D − .−1/ ηj.s − 1/ ; jsj < 3; (14) 1 − s s n ζ . / nD0 jDn D P1 − m − m where 1=s mD0. 1/ .s 1/ . For the digamma term in Eq. (13) we obtain [17] 1 1 X .1 − s/ D −γ C − ζ .k C 1/.s − 1/k: (15) s − 1 kD1 There are several means by which to expand the cotangent term in Eq. (13). We could use the trigonometric definitions, and obtain the needed Taylor series coefficients at s D 1 by applying the Faa di Bruno formula for the nth derivative of a composite function directly to the cotangent function, or to its − csc2 derivative. However, we avoid this method and invoke polygamma functions. From the reflection formula satisfied by the digamma function, we have π π 1 h z z i cot z D 1 − − ; (16) 2 2 2 2 2 which immediately delivers π d n π 1 h z z i cot z D .−1/n .n/ 1 − − .n/ ; (17) 2 dz 2 2nC1 2 2 .j/ .n/ nC1 nC1 where is the polygamma function.
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