Geodesic
Distribution of~prime numbers and the discrete spectrum of~the Laplace operator
Izvestiya. Mathematics , Tome 84 (2020) no. 5, pp. 960-977.

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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. 
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D. A. Popov. Distribution of~prime numbers and the discrete spectrum of~the Laplace operator. Izvestiya. Mathematics , Tome 84 (2020) no. 5, pp. 960-977. http://geodesic.mathdoc.fr/item/IM2_2020_84_5_a5/

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