Plana's summation formula as a modular relation and applications

Abstract

In this paper, we shall locate Plana's summation formula à la Koshlyakov as a form of the modular relation for the Riemann zeta-function and an analytic continuation technique furnished by Lemma 1. Then by Plana's summation formula, we shall prove an important integral representation for the Hurwitz–Lerch zeta-function through confluent hypergeometric functions, which cover a wide range of integral representations attributed to famous mathematicians. We shall also locate the genesis of Mikolás’ formula and prove that the functional equations for the Hurwitz zeta-function and the Riemann zeta-function are equivalent.