Abstract
Contour integral representations of Riemann's Zeta function and Dirichlet's Eta (alternating Zeta) function are presented and investigated. These representations flow naturally from methods developed in the 1800s, but somehow they do not appear in the standard reference summaries, textbooks, or literature. Using these representations as a basis, alternate derivations of known series and integral representations for the Zeta and Eta function are obtained on a unified basis that differs from the textbook approach, and results are developed that appear to be new.
Highlights
Riemann’s Zeta function ζ(s) and its sibling Dirichlet’s function η(s) play an important role in physics, complex analysis, and number theory and have been studied extensively for several centuries
The general importance of a contour integral representation of any function has been known for almost two centuries, so it is surprising that contour integral representations for both ζ(s) and η(s) exist that cannot be found in any of the modern handbooks (NIST, [1, Section 25.5(iii)]; Abramowitz and Stegun, [2, Chapter 23]), textbooks (Apostol, [3, Chapter 12]; Olver, [4, Chapter 8.2]; Titchmarsh, [5, Chapter 4]; Whittaker and Watson, [6, Section (13.13)]), summaries (Edwards, [7], Ivic, [8, Chapter 4]; Patterson, [9]; Srivastava and Choi, [10, 11]), compendia (Erdelyi et al, [12, Section 1.12] and Chapter 17), tables (Gradshteyn and Ryzhik, [13], 9.512; Prudnikov et al, [14, Appendix II.7]), and websites [1, 15,16,17,18,19] that summarize what is known about these functions
Many of the results obtained are disparate and difficult to categorize in a unified manner but share the common theme that they are all somehow obtained from a study of the revived integral representations
Summary
Riemann’s Zeta function ζ(s) and its sibling Dirichlet’s (alternating zeta) function η(s) play an important role in physics, complex analysis, and number theory and have been studied extensively for several centuries. (Recently, Srivastava and Choi [11, pages 169–172], have reproduced Lindelof ’s analysis, failing to arrive explicitly at (7); they do write a general form of (8) ([11, Section 2.3, Equation (38)]), it is studied only for the case c = 1/2.) (Note added in proof: after a draft copy of this paper was posted, I was made aware that Ruijsenaars has previously obtained a generalized form of (7) for the Hurwitz zeta function ([25, Equation (4.22)]) He notes (private communication) that he could not find prior references to his result in the literature).
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