Involve: A Journal of Mathematics

• Involve
• Volume 1, Number 1 (2008), 111-121.

An asymptotic for the representation of integers as sums of triangular numbers

Abstract

Motivated by the result of Rankin for representations of integers as sums of squares, we use a decomposition of a modular form into a particular Eisenstein series and a cusp form to show that the number of ways of representing a positive integer $n$ as the sum of $k$ triangular numbers is asymptotically equivalent to the modified divisor function $σ2k−1♯(2n+k)$.

Article information

Source
Involve, Volume 1, Number 1 (2008), 111-121.

Dates
Accepted: 19 January 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.involve/1513799074

Digital Object Identifier
doi:10.2140/involve.2008.1.111

Mathematical Reviews number (MathSciNet)
MR2403070

Zentralblatt MATH identifier
1229.11066

Subjects
Primary: 11F11: Holomorphic modular forms of integral weight

Citation

Atanasov, Atanas; Bellovin, Rebecca; Loughman-Pawelko, Ivan; Peskin, Laura; Potash, Eric. An asymptotic for the representation of integers as sums of triangular numbers. Involve 1 (2008), no. 1, 111--121. doi:10.2140/involve.2008.1.111. https://projecteuclid.org/euclid.involve/1513799074

References

• T. M. Apostol, Modular functions and Dirichlet series in number theory, 2nd ed., Graduate Texts in Mathematics 41, Springer, New York, 1990.
• N. Koblitz, Introduction to elliptic curves and modular forms, 2nd ed., Graduate Texts in Mathematics 97, Springer, New York, 1993.
• K. Ono, S. Robins, and P. T. Wahl, “On the representation of integers as sums of triangular numbers”, Aequationes Math. 50:1-2 (1995), 73–94.
• R. A. Rankin, “Sums of squares and cusp forms”, Amer. J. Math. 87 (1965), 857–860.