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A segment of Euler product associated to a certain Dirichlet series

Rajat Gupta, Aditi Savalia

Abstract

In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function $σ_α(n):=\sum_{d|n}d^α$. We obtain an exact identity relating the Dirichlet series $ζ(s)ζ(s-α)$ and a segment of the Euler product attached to it. Specifically, our main theorems are valid in the critical strip.

A segment of Euler product associated to a certain Dirichlet series

Abstract

In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function . We obtain an exact identity relating the Dirichlet series and a segment of the Euler product attached to it. Specifically, our main theorems are valid in the critical strip.

Paper Structure

This paper contains 8 sections, 11 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Theorems and Intermediate results
  4. Proofs of Lemmas, main theorem and their corollaries
  5. Proof of Lemma \ref{['Mellin 1.0']}
  6. Proofs of Theorems \ref{['doublesumBessel']} and \ref{['Maintheorem1']}
  7. Proofs of Theorem \ref{['Theoalpha0eqnintro']} and Corollary \ref{['coroalpha0>1']}
  8. Acknowledgment

Key Result

Theorem 1.1

Let $\mathbb{P}\neq\emptyset$, $\Gamma(s)$ be the gamma function and denote $(n,\bar{\mathbb{P}})$ to be the greatest common divisor of $n$ and $\bar{\mathbb{P}}$. Then for complex numbers $s$ and $\tau$, where Re$(s)>0$Here we have corrected the condition from any complex $s$ to Re$(s)>0$ which Lav

Theorems & Definitions (18)

  • Theorem 1.1: Lavrik, lavrik5
  • Theorem 1.2
  • Theorem 1.3: B.C. Berndt, Y. Lee, and J. Sohn berndtleesohn
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Theorem 3.1
  • ...and 8 more