Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-14T02:30:30.188Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic identities for additive convolutions of sums of divisors

Published online by Cambridge University Press:  01 April 2022

ROBERT J. LEMKE OLIVER ,
SUNROSE T. SHRESTHA  and
FRANK THORNE
ROBERT J. LEMKE OLIVER
Affiliation:
Department of Mathematics, Tufts University, 503 Boston Ave, Medford, MA 02155, U.S.A. e-mail: [email protected]
SUNROSE T. SHRESTHA
Affiliation:
Department of Mathematics and Computer Science, Wesleyan University, 45 Wyllys Ave, Middletown, CT 06459, U.S.A. e-mail: [email protected]
FRANK THORNE
Affiliation:
Department of Mathematics, University of South Carolina, 1523 Greene St, Columbia, SC 29201, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 1 , January 2023 , pp. 59 - 78
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Apostol, T. M.. Introduction to Analytic Number Theory (Springer-Verlag, New York-Heidelberg, 1976). Undergraduate Texts in Mathematics.Google Scholar
Delecroix, V., Goujard, E., Zograf, P. and Zorich, A.. Contribution of one-cylinder square-tiled surfaces to Masur–Veech volumes. Astérisque (415, Quelques aspects de la théorie des systèmes dynamiques: un hommage à Jean-Christophe Yoccoz. I) (2020), 223–274.CrossRefGoogle Scholar
Deshouillers, J.-M. and Iwaniec, H.. Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70(2) (1982/83), 219–288.CrossRefGoogle Scholar
Eskin, A. and Okounkov, A.. Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Invent. Math. 145 (2001), 59–103.CrossRefGoogle Scholar
Gradshteyn, I. S. and Ryzhik, I. M.. Table of integrals, series, and products. Fourth edition prepared by Ju. V. Geronimus and M. Ju. Cetlin. Translated from the Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey (Academic Press, New York-London, 1965).Google Scholar
Halberstam, H.. Four asymptotic formulae in the theory of numbers. J. London Math. Soc. 24 (1949), 13–21.Google Scholar
Halberstam, H.. An asymptotic formula in the theory of numbers. Trans. Amer. Math. Soc. 84 (1957), 338–351.CrossRefGoogle Scholar
Hubert, P. and Lelievre, S.. Prime arithmetic teichmüller discs in $\mathcal{H}(2)$ . Israel J. Math. 151(1) (2006), 281–321.CrossRefGoogle Scholar
Ingham, A. E.. Some asymptotic formulae in the theory of numbers. J. London Math. Soc. 2(3) (1927), 202–208.Google Scholar
Kontsevich, M. and Zorich, A.. Connected components of the moduli spaces of abelian differentials with prescribed singularities. Invent. Math. 153 (2003), 631–678, 2003.CrossRefGoogle Scholar
Kuznetsov, N. V.. Convolution of Fourier coefficients of Eisenstein–Maass series. volume 129, pages 43–84 (1983). Automorphic functions and number theory, I.Google Scholar
Lelievre, S. and Royer, E.. Orbit countings in $\mathcal{H}(2)$ and quasimodular forms. Internat. Math. Res. Not. 2006(42151) (2006), 1–30.Google Scholar
Masur, H.. Interval exchange transformations and measured foliations. Ann. Math. 115 (1982), 169–200.CrossRefGoogle Scholar
Masur, H.. Ergodic theory of translation surfaces. In B. Hasselblatt and A. Katok, editors, Handbook of dynamical sytems, volume 1B, pages 527–547 (Elsevier B. V., 2006).CrossRefGoogle Scholar
McMullen, C. T.. Teichmüller curves in genus two: discriminant and spin. Math. Ann. 333(1) (2005), 87–130.CrossRefGoogle Scholar
Miyagawa, T.. Approximate functional equations for the Hurwitz and Lerch zeta-functions. Comment. Math. Univ. St. Pauli 66(1–2) (2017), 15–27.Google Scholar
Motohashi, Y.. The binary additive divisor problem. Ann. Sci. école Norm. Sup. (4) 27(5) (1994), 529–572.CrossRefGoogle Scholar
Ramanujan, S.. On certain arithmetical functions Trans. Camb. Phil. Soc. 22 (1916), no. 9, 159–184]. In Collected papers of Srinivasa Ramanujan, pages 136–162 (AMS Chelsea Publ., Providence, RI, 2000).Google Scholar
Selberg, A.. On the estimation of Fourier coefficients of modular forms. In Proc. Sympos. Pure Math., Vol. VIII, pages 1–15 (Amer. Math. Soc., Providence, R.I., 1965).CrossRefGoogle Scholar
Shparlinski, I. E. and Zhang, T.. Cancellations amongst Kloosterman sums. Acta Arith. 176(3) (2016), 201–210.CrossRefGoogle Scholar
Shrestha, S. T.. Counting formulae for square-tiled surfaces in genus two. Ann. Math. Blaise Pascal 27(1) (2020), 83–123.CrossRefGoogle Scholar
Veech, W.. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. 115(2) (1982), 201–242.CrossRefGoogle Scholar
Veech, W.. Moduli spaces of quadratic differentials. J. Anal. Math. 55(1) (1990), 117–171.CrossRefGoogle Scholar
Zmiaikou, D.. The probability of generating the symmetric group with a commutator condition. Preprint (2012), available at https://arxiv.org/abs/1205.6718.Google Scholar
Zorich, A.. Square tiled surfaces and Teichmüller volumes of the moduli spaces of abelian differentials. In Rigidity in Dynamics and Geometry (Cambridge, 2000), pages 459–471 (Springer, Berlin, 2002).CrossRefGoogle Scholar