Certain relationships among polygamma functions, Riemann zeta function and generalized zeta function

Junesang Choi,Chao-Ping Chen

doi:10.1186/1029-242x-2013-75

Junesang Choi, Chao-Ping Chen

Open Access

https://doi.org/10.1186/1029-242x-2013-75

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Abstract

Many useful and interesting properties, identities, and relations for the Riemann zeta function and the Hurwitz zeta function have been developed. Here, we aim at giving certain (presumably) new and (potentially) useful relationships among polygamma functions, Riemann zeta function, and generalized zeta function by modifying Chen’s method. We also present a double inequality approximating by a more rapidly convergent series. MSC:11M06, 33B15, 40A05, 26D07.

Highlights

The Riemann zeta function ζ (s) is defined by ∞ζ (s) := ns (s) > . ( . ) n=The Hurwitz zeta function ζ (s, a) is defined by ζ (s, a) := (k + a)–s (s) > ; a ∈ C \ Z, k=where C and Z– denote the sets of complex numbers and nonpositive integers, respectively
The number-theoretic properties of ζ (s) are exhibited by the following result known as Euler’s formula, which gives a relationship between the set of primes and the set of positive integers: ζ (s) = – p–s – (s) >, ( . )
2 Main results We first prove a relationship between polygamma functions and Riemann zeta functions asserted by Theorem

Summary

Introduction

The Riemann zeta function ζ (s) is defined by ∞ζ (s) := ns (s) > . ( . ) n=The Hurwitz (or generalized) zeta function ζ (s, a) is defined by ζ (s, a) := (k + a)–s (s) > ; a ∈ C \ Z– , k=where C and Z– denote the sets of complex numbers and nonpositive integers, respectively. The Hurwitz (or generalized) zeta function ζ (s, a) is defined by Is a well-known (useful) relationship between the polygamma functions ψ(n)(s) and the generalized zeta function ζ (s, a): ψ (n)(s) = (– )n+ n!

Results

Conclusion