Brundaban Sahu

EVALUATION OF THE CONVOLUTION SUMS ∑l+15m=nσ(l)σ(m) AND ∑3l+5m=nσ(l)σ(m) AND AN APPLICATION

We evaluate the convolution sums ∑l,m∈ℕ,l+15m=nσ(l)σ(m) and ∑l,m∈ℕ,3l+5m=nσ(l)σ(m) for all n ∈ ℕ using the theory of quasimodular forms and use these convolution sums to determine the number of representations of a positive integer n by the form We also determine the number of representations of positive integers by the quadratic form by using the convolution sums obtained earlier by Alaca, Alaca and Williams [Evaluation of the convolution sums ∑l+6m=nσ(l)σ(m) and ∑2l+3m=nσ(l)σ(m), J. Number Theory124(2) (2007) 491–510; Evaluation of the convolution sums ∑l+24m=nσ(l)σ(m) and ∑3l+8m=nσ(l)σ(m), Math. J. Okayama Univ.49 (2007) 93–111].

Supercongruences for sporadic sequences

We prove two-term supercongruences for generalizations of recently discovered sporadic sequences of Cooper. We also discuss re- cent progress and future directions concerning other types of supercon- gruences.

Congruences via modular forms

We prove two congruences for the coecients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide tables of congruences for numbers which appear in similar power series expansions and in the study of integral solutions of Ap ery-like dierential equations.

On the number of representations of an integer by certain quadratic forms in sixteen variables

We evaluate the convolution sums ∑l,m∈ℕ,l+2m=n σ3(l)σ3(m), ∑l,m∈ℕ,l+3m=n σ3(l) × σ3(m), ∑l,m∈ℕ,2l+3m=n σ3(l)σ3(m) and ∑l,m∈ℕ,l+6m=n σ3(l)σ3(m) for all n ∈ ℕ using the theory of modular forms and use these convolution sums to determine the number of representations of a positive integer n by the quadratic forms Q8 ⊕ Q8 and Q8 ⊕ 2Q8, where the quadratic form Q8 is given by

On the number of representations by certain octonary quadratic forms with coefficients 1, 2, 3, 4 and 6

In this paper, we find formulas for the number of representations of certain diagonal octonary quadratic forms with coefficients

On the Representations of a Positive Integer by Certain Classes of Quadratic Forms in Eight Variables

1,2,3,4

Representations of an Integer by Some Quaternary and Octonary Quadratic Forms

and

Rankin's method and Jacobi forms of several variables

6

On the Fourier expansions of Jacobi forms of half-integral weight

. We obtain these formulas by constructing explicit bases of the space of modular forms of weight

Supercongruences for Apéry-like numbers

4