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On logarithmic asymptotics for the number of restricted partitions in the exponential case

Murat V. Vakhitov and Dmitrii S. Minenkov
Vol. 12 (2023), No. 4, 297–309
DOI: 10.2140/moscow.2023.12.297
Abstract

Asymptotics for the logarithm of the number of restricted partitions is calculated in the case of exponential growth of the counting function. In terms of abstract number theory this problem is the inverse problem of the distribution of abstract primes, i.e., for a given counting function of abstract primes we calculate the asymptotics of abstract integers. While calculating the number of integers, only the limited number of abstract primes is considered and the constructed asymptotics is uniform with respect to this number. We consider only the leading term of logarithmic asymptotics, which is obtained using elementary estimations. The considered case includes the case of integers and primes themselves.

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Keywords
restricted partitions, abstract primes, abstract integers, counting function, asymptotics
Mathematical Subject Classification
Primary: 11P70, 11P81
Milestones
Received: 27 May 2022
Revised: 25 November 2023
Accepted: 12 December 2023
Published: 5 January 2024
Authors
Murat V. Vakhitov
Department of Mechanics and Mathematics
Lomonosov Moscow State University
Moscow
Russia		 National Research University Higher School of Economics
Moscow
Russia
Dmitrii S. Minenkov
Ishlinsky Institute for Problems in Mechanics
Moscow
Russia		 National Research University Higher School of Economics
Moscow
Russia
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