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EVALUATION OF CONVOLUTION SUMS $$\sum\limits_{l + km = n} {\sigma (l)\sigma (m)} $$ AND $$\sum\limits_{al + bm = n} {\sigma (l)\sigma (m)} $$ FOR k = a · b = 21, 33, AND 35

Part of: Forms and linear algebraic groups Zeta and $L$-functions: analytic theory Elementary number theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  16 July 2021

K. PUSHPA  and
K. R. VASUKI
K. PUSHPA
Affiliation:
Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysuru 570 006, India e-mails: [email protected]; [email protected]
K. R. VASUKI
Affiliation:
Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysuru 570 006, India e-mails: [email protected]; [email protected]
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Abstract

The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$, for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

MSC classification

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Secondary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 11E25: Sums of squares and representations by other particular quadratic forms 11F20: Dedekind eta function, Dedekind sums
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 2 , May 2022 , pp. 434 - 453
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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