Narrow Results

The evaluation of the sum ∑_m<n/9σ(m)σ(n - 9m) is carried out for all positive integers n. This evaluation is used to detemine the number of solutions to

We use the recent evaluation of certain convolution sums involving the sum of divisors function to determine the number of representations of a positive integer by certain diagonal octonary quadratic forms whose coefficients are 1, 2 or 4.

In this study we use some known convolution sums to find the representation number for each of the three octonary quadratic forms , and .

We evaluate the convolution sums ∑_{l,m∈ℕ,l+15m=n}σ(l)σ(m) and ∑_{l,m∈ℕ,3l+5m=n}σ(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

In this study we derive formulae for the number of representations of n by the octonary quadratic forms for any n ∈ ℕ₀.

Let N(a₁, …, a₄; n) denote the number of representations of an integer n by the form . In this paper we derive formulae for N(1, 1, 1, 2; n) and N(1, 2, 2, 2; n). These formulae are given in terms of σ₃(n).

In this paper, the convolution sums ∑_i+3j=nσ(i)σ₃(j) and ∑_3i+j=nσ(i)σ₃(j) are evaluated for all n ∈ ℕ, which answers an open problem given by Huard, Ou, Spearman and Williams. As an application, the number of representations of a positive integer n by the form

The theory of quasimodular forms is used to evaluate the convolution sums

In this paper, the convolution sum ∑_i+25j=n σ(i)σ(j) is evaluated for all n ∈ ℕ based on some identities due to Berndt, Lemire and Williams.

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

In this work, we evaluate the convolutions ∑_l+36m=n σ(l)σ(m) and ∑_4l+9m=n σ(l)σ(m) for all positive n. As an application, these evaluations are used to determine the number of representations of a positive integer by the forms and .

We determine the convolution sums ${\sum }_{l+27m=n}\sigma (l)\sigma (m)$ and ${\sum }_{l+32m=n}\sigma (l)\sigma (m)$ for all positive integers $n$. We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of $n$ by the octonary quadratic forms $x_1^2+x_1{x}_2+x_2^2+x_3^2+x_3{x}_4+x_4^2+9(x_5^2+x_5{x}_6+x_6^2+x_7^2+x_7{x}_8+x_8^2)$ and $x_1^2+x_2^2+x_3^2+x_4^2+8(x_5^2+x_6^2+x_7^2+x_8^2)$. A modular form approach is used.

We express products of Lambert series as power series. We use these Lambert series-to-power series identities to obtain new Liouville identities with two functions. Many of the known Liouville identities follow from our new identities. We explore the relationships between Lambert series, new Liouville identities, and convolution sums. We then obtain recursive formulae for various convolution sums.

In this paper, the convolution sums ${\sum }_{i+12j=n}\sigma (l){\sigma }_3(m)$, ${\sum }_{3i+4j=n}\sigma (l){\sigma }_3(m),$${\sum }_{4i+3j=n}\sigma (l){\sigma }_3(m)$ and ${\sum }_{12i+j=n}\sigma (l){\sigma }_3(m)$ are evaluated for all $n\in \mathbb{N},$ and their evaluations are used to determine the number of representation of a positive integer $n$ by the forms $x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2+3x_9^2+3x_{10}^2+3x_{11}^2+3x_{12}^2$ and $x_1^2+x_2^2+x_3^2+x_4^2+3x_5^2+3x_6^2+3x_7^2+3x_8^2+3x_9^2+3x_{10}^2+3x_{11}^2+3x_{12}^2.$

In this paper, we provide two identities about binomial convolution sums of ${\sigma }_r^{♭}(n;N/4,N)$ with $N/4\in \mathbb{N}$, which are expressed in terms of Euler and Bernoulli polynomials. A recent result of Kim, Bayad and Park turns out to be a special case of one of the two identities when $N=4$.

Recently, many identities for the convolution sum $W_{a,b}(n):={\sum }_{a\ell +bm=n}\sigma (\ell )\sigma (m)$ of the divisor function $\sigma (n):={\sum }_{d|n}d$ have been obtained since Royer obtained by the theory of quasimodular forms. We also present new identities for $ab=17$, $29$, $41$, $47$, $59$ and $71$ by using quasimodular forms.