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A Series Representation for Riemann's Zeta Function and Some Journal of Classical Analysis Volume 17, Number 2 (2021), 129–167 doi:10.7153/jca-2021-17-09 A SERIES REPRESENTATION FOR RIEMANN’S ZETA FUNCTION AND SOME INTERESTING IDENTITIES THAT FOLLOW MICHAEL MILGRAM Abstract. Using Cauchy’s Integral Theorem as a basis, what may be a new series representation for Dirichlet’s function η(s), and hence Riemann’s function ζ(s), is obtained in terms of the Exponential Integral function Es(iκ) of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are un- earthed. The incomplete functions ζ ±(s) and η±(s) are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths be- tween the endpoints. 1. Introduction In a recent paper [1], I have developed an integral equation for Riemann’s func- tion ξ (s), based on LeClair’s series [2] involving the Generalized Exponential Integral. Glasser, in a subsequent paper [3], has shown that this integral equation is equivalent to the standard statement of Cauchy’s Integral Theorem applied to ξ (s), from which he closed the circle [3, Eq. (13)] by re-deriving LeClair’s original series representation. This raises the question of what will be found if Cauchy’s Integral Theorem is applied to other functions related to Riemann’s function ζ(s), notably the so-called Dirichlet function η(s) (sometimes called the alternating zeta function), working backwards to a series representation? That question is the motivation for this work. Section 2 gives some preliminaries and develops the reduction of Cauchy’s Inte- gral Theorem to yield a series representation of η(s) in terms of (generalized) Integro- Exponential functions [4] Es(κ) (a.k.a Incomplete Gamma functions – see (4.1)). Al- though it is well-known that ζ(s) can be expressed [2], [5] in terms of such functionsin various ways, (see Appendix B) the form derived here differs substantially from these since it involves a series that indexes half-integer values of the imaginary argument of Es(iκ) and does not possess generic s (1 s) symmetry. Several subsections explore variations of this series, revealing some↔ interest− ing identities, not all of which pertain to either ζ(s) or η(s). Section 3 presents a brief analysis of these results as they pertain to ζ ′(s). In Section 4, the series representation is converted into an integral Mathematics subject classification (2010): 11M06, 11M26, 11M35, 11M99, 26A09, 30B40, 30E20, 33C20, 33B20, 33B99. Keywords and phrases: Riemann zeta function, Dirichlet eta function, alternating zeta function, incom- plete zeta function, generalized exponential integral, Bernoulli numbers, Euler numbers, harmonic numbers, infinite series, evaluation of integrals. Ð c D , Zagreb 129 Paper JCA-17-09 130 M. MILGRAM representation whose endpoints are complex, and in a number of subsections this repre- sentation is reduced to a real integral representation by choosing a number of different paths between the endpoints in the complex plane, yielding an evaluation of several unusual integrals. Appendix (C) collects some known results referenced in the main text, Appendix (D) contains the proof of a needed theorem and Appendix (E) presents the details of a lengthy derivation required in Subsection (2.4). In the late 1800’s, Russell ([6]) initiated a tradition by presenting a list of 12 in- tegral identities with little in the way of proof. This tradition has been recently (2008) reinvigorated by Amdeberhan and Moll([7]); more recently (2018) Coffey ([8]) has pro- vided some proofs and generalizations of Russell’s identities. Continuing the tradition, various subsections throughout this work present (special case) integrals that satisfy Russell’s presumed criterion – that being that they have no apparent application other than that they are interesting and involve complicated trigonometric and hyperbolic functions. See also [9]. Therefore, a (baker’s) dozen of the most appealing of the interesting results ob- tained herein follows immediately below, cross-referenced to where each appears in the text of this paper. Departing from Russell’s approach, each cross-reference to the later text, provides sufficient information to allow an informed reader to obtain a formal proof if (s)he so desires. A list of notations appears in Appendix A. The main result upon which this paper is based (see (2.24)): • (n 1)/2 E s s 1 ⌊ − ⌋ Γ(2 j + 1 s) (2 j) (2 1)ζ (s)= π − sin(πs/2) ∑ − − − j=0 Γ(2 j + 1) s n iπ n/2 ∞ n π − e E1+n s ( iπ (k + 1/2))+ ( 1) E1+n s (iπ (k + 1/2)) + − − − − 2Γ(s n)cos(πs/2) ∑ (k + 1/2)n − k=0 The sum of an infinite series (see (2.35)): • p 2m m iπ p/2 ∞ p π − ( 1) e− ( 1) E p (iπ (k + 1/2)) + E p ( iπ (k + 1/2)) − − − ∑ − p+2m − Γ(1 + p) k=0 (k + 1/2)− m E (2 j) = . (1.1) ∑ m j j − j= p/2+m+1/2 Γ(2 2 + 1)Γ(2 + 1) ⌊− ⌋ − ζ(2m + 1) expressed in terms of E1(κ) (see (2.39)): • m E (2 j)ψ (1 2 j + 2m) 22m+1 1 ζ (2m + 1)= ( 1)m π2m − − − − ∑ Γ(2 j + 1)Γ(1 2 j + 2m) j=0 − ∞ i E1 (iπ (k + 1/2)) E1 ( iπ (k + 1/2)) + ∑ − 2m−+1 . π k=0 (k + 1/2) A SERIES REPRESENTATION FOR RIEMANN’S ZETA FUNCTION 131 Bernoulli/Euler and Harmonic numbers are related (see (2.46)): • m 1 m 2k 2k − E (2 j)H2m 2 j 2 2 1 B2k − = − ∑ Γ(1 2 j + 2m)Γ(2 j + 1) ∑ (1 2k + 2m)Γ(2 2k + 2m)Γ(2k + 1) j=0 − k=1 − − 1 − 2mΓ(1 + 2m) An integral involving Euler polynomials (see (2.49)): • 2m 1 m 1 2 E (2m,u/2) − E (2 j)H2m 2 j du = ∑ − m > 0, Γ(2m + 1) 0 u − Γ(2 j + 1)Γ(2m 2 j + 1) Z j=0 − For integer p > m, m < cm < m+1, a contour integral representation for ζ(2m+1) • (see (E.16)): 22m+1 1 ζ (2m + 1) −p i( 1) cm+i∞ Γ(2p 2t)π2t ∞ k k 2t 2 p 2m 1 t = −2p+1 − ∑ ( 1) ( + 1/2) − − − d (1.2) π cm i∞ sin(π t) − Z − k=0 p 2 m m − − Γ(2 j+2)E (2 j+2+2m) E (2 j)ψ (1 2 j+2m) ( 1)m π2m + − . − − ∑ Γ(2 j+3+2m) ∑ Γ(1 2 j+2m)Γ(2 j+1) j=0 j=0 − ! A divergent, asymptotic series representation for the alternating Hurwitz Zeta Func- • tion η(1,(t + 1)/2) (see (2.64)): 2m E (2k) t t m > η(1,( + 1)/2) ∑ 2k+1 ( ∞), 0. (1.3) ∼ k=0 t → ∀ A representation for the lower incomplete Zeta function ζ (s) (see (4.10)): • − s 1 π/2 (π/2) sin(θ) 2 − e− sin((π/2) cos(θ)+ θ (s 1)) ζ − (s)= 1 s − dθ 2 − 1 π/2 cosh(π sin(θ)) cos(π cos(θ)) − Z− − An interesting integral with 0 6 2n (see (4.17)): • π/2 (π/2) sin(θ) 1 2n sin((π/2) cos(θ) + (2n 1)θ)e− n 2n 2 − 1 − dθ = ( 1) B2n π − π/2 cosh(π sin(θ)) cos(π cos(θ)) − Γ(2n + 1) Z− − An interesting integral with 2n > 0 (see (4.19)): • π/2 sin((π/2) cos(θ) (2n + 1)θ)e (π/2) sin(θ) − − dθ = 0 π/2 cosh(π sin(θ)) cos(π cos(θ)) Z− − 132 M. MILGRAM An integral interesting on its own merits (see (4.28)): • π/2 cos(θ)sin((π/2) cos(θ))e (π/2) sin(θ) − dθ = 1/2 + π2/48, (1.4) π/2 cosh(π sin(θ)) cos(π cos(θ)) Z− − A theorem for p > 0, (see (D.2)): • p E2p ( iπ (k + 1/2)) + E2p (iπ (k + 1/2)) ( 1) Γ(2 p) − 2 p 1 − (π (k + 1/2)) − p 1 j k − ( 1) Γ(2 j) = 2 ( 1) ∑ − 2 j 2(Si(π(k + 1/2)) π/2) − j=1 (π (k + 1/2)) − − A hypergeometric identity (see (D.10)): • 1 1 3 π2 (k + 1/2)2 1F2 p; , p; 2 − 2 2 − − 4 ( 1)p+1 (π (k + 1/2))2 p 1 3 3 π2 (k + 1/2)2 = − F ; , ; − Γ(2 p 1) 1 2 2 2 2 4 − k+p 2 p 1p 1 j ( 1) (π (k + 1/2)) − − Γ(2 j)( 1) + − ∑ − Γ(2 p 1) (π (k + 1/2))2 j − j=1 2. A series representation for ζ(s) By way of review, the standard definition for Dirichlet’s function η(s) (also called the alternating zeta function) is given by ∞ k s η(s)= ∑ ( 1) k− (2.1) k=1 − valid for all s, related to ζ(s) by 1 s η(s) = (1 2 − )ζ(s). (2.2) − Another well-known representation is ∞ (2s 1)ζ(s)= ∑ 1/(k + 1/2)s , ℜ(s) > 1. (2.3) − k=0 1 s The Cauchy Integral Theorem for η(s), with an additional factor 2 − embedded for convenience, is 1 v 1 v 1 s 1 s 1 1 2 − ζ (v)2 − 2 − 1 2 − ζ (s)= − dv (2.4) − 2πi s v I − A SERIES REPRESENTATION FOR RIEMANN’S ZETA FUNCTION 133 where the contour of integration encloses the non-negative real axis as well as the pole at the point v = s in a clockwise direction.
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