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Identities for the product of Two Dirichlet Series Satisfying Hecke's Functional Equation

Bruce C. Berndt, Likun Xie

TL;DR

The paper addresses the problem of expressing the product of two Dirichlet series that satisfy Hecke-type functional equations. It develops a general product formula by combining a Perron-type Riesz-sum framework with contour-shift analysis and Meijer $G$-function representations, yielding an identity for $\varphi(u)\psi(v)$ in terms of pole data and derivatives of Meijer-$G$-based auxiliary functions. The authors validate the method by deriving explicit arithmetical identities for classical series, including the Ramanujan $\tau$-function, the Riemann zeta function, and the $l$-th divisor function, thereby illustrating how the framework encodes known and new product relations. They also compare their approach with prior work, clarifying several gaps and errors in earlier attempts and highlighting the careful analytic conditions needed for termwise manipulations. The results provide a versatile toolkit for obtaining approximate functional equations for products of Dirichlet series and for generating concrete identities in arithmetic contexts.

Abstract

We derive a general formula for the product of two Dirichlet series that satisfy Hecke's functional equation. Several examples are provided to demonstrate the applicability of the formula. In addition, we discuss prior work on similar products and clarify certain issues arising in the existing literature.

Identities for the product of Two Dirichlet Series Satisfying Hecke's Functional Equation

TL;DR

The paper addresses the problem of expressing the product of two Dirichlet series that satisfy Hecke-type functional equations. It develops a general product formula by combining a Perron-type Riesz-sum framework with contour-shift analysis and Meijer -function representations, yielding an identity for in terms of pole data and derivatives of Meijer--based auxiliary functions. The authors validate the method by deriving explicit arithmetical identities for classical series, including the Ramanujan -function, the Riemann zeta function, and the -th divisor function, thereby illustrating how the framework encodes known and new product relations. They also compare their approach with prior work, clarifying several gaps and errors in earlier attempts and highlighting the careful analytic conditions needed for termwise manipulations. The results provide a versatile toolkit for obtaining approximate functional equations for products of Dirichlet series and for generating concrete identities in arithmetic contexts.

Abstract

We derive a general formula for the product of two Dirichlet series that satisfy Hecke's functional equation. Several examples are provided to demonstrate the applicability of the formula. In addition, we discuss prior work on similar products and clarify certain issues arising in the existing literature.

Paper Structure

This paper contains 11 sections, 6 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Background Set-up
  3. Dirichlet Series with a General Functional Equation
  4. Meijer $G$-Function
  5. Integral with Riesz Sum
  6. A Formula for the Product $\varphi(u)\psi(v)$
  7. Arithmetical Identities
  8. Dirichlet Series for the Ramanujan $\tau$-function
  9. The Riemann Zeta Function
  10. The $l$-th Divisor Function
  11. Additional Examples

Key Result

Theorem 1.1

(Approximate Functional Equation for $\zeta(s)$) Let $x$, $y$, $h$, and $k$ be positive real numbers such that Then, as $|t| \to \infty$,

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: functional; see also means
  • Lemma 2.2
  • Theorem 3.1
  • proof