A series representation for Riemann's zeta function and some interesting identities that follow

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 E s (iκ) of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed.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 between the endpoints. Mathematics subject classification (2010): 11M06,

An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution

“…Note that the left and right limits of each of the horizontal "treads" are open and that the values at the midpoints of the "risers" are obtained by both an analytic and a numerical evaluation of the integral, not by decree. in agreement with [1,Eq. (4.13)].…”

“…As noted in [1], by pairing integrals living inside the critical strip 0 ≤ σ ≤ 1 with companions that are tractable and live outside, it becomes possible to evaluate the companion integral Z(1/2, r) by applying either the Master Theorem or analytic continuation and compare with previous results. From (2.2),(2.6) and (2.7) we find…”

“…In a previous work [1], relationships between different parametric instances of inverse Mellin transform integrals of the form…”

“…as one tine of the Dirac comb function when considered only as a function of the variable r. 1 Remark 1.1.1. For a proof of (1.7), see Section 3.1.…”

“…For a proof of (1.7), see Section 3.1. 1 The Dirac comb function is the set of Dirac delta functions of unit separation.…”