Work

An explicit formula for the zeros of the Riemann zeta function and the statistics of their local distribution

Year: 2025

Type: article

Abstract: We present an explicit formula for a weighted sum over the zeros of the Riemann zeta function. This weighted sum is evaluated in terms of a sum over the prime numbers, weighted with the help of the Hermite polynomials. Then we apply this result to deduce, under the Riemann hypothesis, a formula describing the local distribution of zeros of the Riemann zeta function lying along the critical line. For this application we make use of the Prime Number Theorem in conjunction with Kac’s formula for the distribution of values of trigonometric polynomials.

Source: The Ramanujan Journal

DOI: 10.1007/s11139-025-01297-y

Authors: Eugenio P. Balanzario, Daniel Eduardo Cárdenas Romero

Institutions: Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,990

OpenAlex ID: W7116760418