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Asymptotic formulas for the logarithm of general L-functions at and near s=1

Kohji Matsumoto, Yumiko Umegaki

Abstract

We consider a general form of L-function L(s) defined by an Euler product and satisfies several analytic assumptions. We show several asymptotic formulas for L(1) and log L(1). In particular those asymptotic formulas are valid for Dirichlet L-functions attached to almost all Dirichlet characters. Our theorems should be compared with former results due to Elliott, Montgomery and Weinberger, etc.

Asymptotic formulas for the logarithm of general L-functions at and near s=1

Abstract

We consider a general form of L-function L(s) defined by an Euler product and satisfies several analytic assumptions. We show several asymptotic formulas for L(1) and log L(1). In particular those asymptotic formulas are valid for Dirichlet L-functions attached to almost all Dirichlet characters. Our theorems should be compared with former results due to Elliott, Montgomery and Weinberger, etc.
Paper Structure (8 sections, 10 theorems, 119 equations)

This paper contains 8 sections, 10 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. General $L$-functions and their truncations
  3. Statement of results
  4. Estimations of logarithmic derivatives
  5. An auxiliary formula for $\log F(\alpha)$
  6. Proof of Theorem \ref{['thm-1']} and Theorem \ref{['thm-2']}
  7. Proof of Theorem \ref{['thm:main_p_s=1']}
  8. Proof of Theorem \ref{['cor:apply_PNT']} and Theorem \ref{['cor:apply_PNT2']}

Key Result

Theorem 1

Let $N\geq 16$ be a positive number. Under Assumptions conti-analy, zero-free and convex, for $7/2 > \delta > 0$ and $A\geq 14/\delta$, we have

Theorems & Definitions (30)

  • Remark 1
  • Theorem 1
  • Corollary 1
  • proof
  • Remark 2
  • Theorem 2
  • Theorem 3
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 20 more