EVALUATION OF CONVOLUTION SUMS AND FOR k = a · b = 21, 33, AND 35

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).

Estimates of shifted convolution sums involving Fourier coefficients of Hecke–Maass eigenform

We obtain certain estimates for averages of shifted convolution sums involving the Fourier coefficients of a normalized Hecke–Maass eigenform and holomorphic cusp form.

Some result for binomial convolution sums of restricted divisor functions

Besge presented the result about the convolution sum of divisor functions. Since then Liouville obtained the generalized version of Besge's formula, which is the binomial convolution sum of divisor functions. In 2004, Hahn obtained the results about the convolution sums of ?d|n(-1)d-1d and ?d|n (-1)n=d-1d. In this paper, we present the results for the binomial con- voltion sums, generalized convolution sums of Hahn, of these divisor functions.

Convolution sums of a divisor function for prime levels

Recently, many identities for the convolution sum [Formula: see text] of the divisor function [Formula: see text] have been obtained since Royer obtained by the theory of quasimodular forms. We also present new identities for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] by using quasimodular forms.