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Application of Meyer's theorem on quasicrystals to exponential polynomials and Dirichlet series

Sergii Favorov

Abstract

A simple necessary and sufficient condition is given for an absolutely convergent Dirichlet series with imaginary exponents and only real zeros to be a finite product of sines. The proof is based on Meyer's theorem on quasicrystals.

Application of Meyer's theorem on quasicrystals to exponential polynomials and Dirichlet series

Abstract

A simple necessary and sufficient condition is given for an absolutely convergent Dirichlet series with imaginary exponents and only real zeros to be a finite product of sines. The proof is based on Meyer's theorem on quasicrystals.

Paper Structure

This paper contains 2 theorems, 38 equations.

Key Result

Theorem 1

Let $Q(z)$ be Dirichlet series S with only real zeros, $h_\gamma$ be coefficients of the Dirichlet series expansion of the function $Q'(z)/Q(z)$ in the half-planes Then the condition is necessary and sufficient for the representation sin.

Theorems & Definitions (2)

  • Theorem
  • Theorem : M, p.26, and KL