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On Berndt's summation formula

Alexander E. Patkowski

Abstract

We offer a proof of a summation formula equivalent to one due to Berndt. Our proof uses the M$\ddot{u}$ntz formula and the Poisson summation formula. By utilizing known properties of Mellin inversion, we give an example from a discontinuous function. Several new applications are offered as corollaries.

On Berndt's summation formula

Abstract

We offer a proof of a summation formula equivalent to one due to Berndt. Our proof uses the Mntz formula and the Poisson summation formula. By utilizing known properties of Mellin inversion, we give an example from a discontinuous function. Several new applications are offered as corollaries.

Paper Structure

This paper contains 3 sections, 4 theorems, 33 equations.

Table of Contents

  1. Introduction and Main Summation formulas
  2. Identity involving the Koshlyakov function
  3. Rearranging Motohashi's formula

Key Result

Theorem 1.1

Assume $f$ satisfies the hypothesis of (1.1) and growth conditions for (1.2). Let $\zeta(s)$ be the Riemann zeta function. Then, assuming absolute convergence, where $\mathfrak{M}^{-1}$ is taken over the vertical line $0<\Re(s)=c<1.$

Theorems & Definitions (8)

  • Theorem 1.1
  • proof
  • Theorem 1.2
  • proof
  • Theorem 1.3
  • proof
  • Theorem 2.1
  • proof