Distribution of prime numbers and the discrete spectrum of the Laplace operator

D. A. Popov

Lomonosov Moscow State University, Belozersky Research Institute of Physico-Chemical Biology

Abstract: We obtain a class of explicit formulae each of which gives an expression for the remainder term in the asymptotic equation for the Chebyshev function in terms of the spectrum of the Laplace operator on the fundamental domain of the modular group.

Keywords: prime numbers, Chebyshev psi-function, spectrum of the Laplace operator, modular group.

UDC: 511.331+515.178.1

MSC: Primary 58J50; Secondary 11M36

Received: 05.12.2019

DOI: 10.4213/im8999

 English version: Izvestiya: Mathematics, 2020, 84:5, 960–977

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