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Asymptotic identities for additive convolutions of sums of divisors.
- Source :
-
Mathematical Proceedings of the Cambridge Philosophical Society . Jan2023, Vol. 174 Issue 1, p59-78. 20p. - Publication Year :
- 2023
-
Abstract
- In a 1916 paper, Ramanujan studied the additive convolution $S_{a, b}(n)$ of sum-of-divisors functions $\sigma_a(n)$ and $\sigma_b(n)$ , and proved an asymptotic formula for it when a and b are positive odd integers. He also conjectured that his asymptotic formula should hold for all positive real a and b. Ramanujan's conjecture was subsequently proved by Ingham, and then by Halberstam with a power saving error term. In this paper, we give a new proof of Ramanujan's conjecture that obtains lower order terms in the asymptotics for most ranges of the parameters. We also describe a connection to a counting problem in geometric topology that was studied in the second author's thesis and which served as our initial motivation in studying this sum. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 03050041
- Volume :
- 174
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Mathematical Proceedings of the Cambridge Philosophical Society
- Publication Type :
- Academic Journal
- Accession number :
- 160865982
- Full Text :
- https://doi.org/10.1017/S0305004122000135