On the generalized Dirichlet beta and Riemann zeta functions and Ramanujan-type formulae for beta and zeta values

TL;DR

This paper introduces generalized Dirichlet beta and Riemann zeta functions using integral definitions involving hyperbolic functions, establishes their functional equations, and derives new Ramanujan-type formulas for zeta values at odd integers.

Contribution

It defines new generalized functions, proves their functional equations, and extends Ramanujan identities to obtain novel formulas for zeta values.

Findings

Established functional equations for the new functions

Derived new Ramanujan-type formulas for zeta at odd integers

Explored consequences of Ramanujan identities

Abstract

We define the generalized Dirichlet beta and Riemann zeta functions in terms of the integrals, involving powers of the hyperbolic secant and cosecant functions. The corresponding functional equations are established. Some consequences of the Ramanujan identity for zeta values at odd integers are investigated and new formulae of the Ramanujan type are obtained.