An Improved Asymptotic on the Representations of Integers as Sums of Products

In this paper, we improve the error terms of Chace’s results in the study by Chace (1994) on the number of ways of writing an integer N as a sum of k products of l factors, valid for k ≥ 3 and l = 2 , 3. More precisely, for l = 2 , 3, we improve the upper bound N k − 1 − 2 k − 2 / k − 1 l + 1 + ε , k ≥ 3 for the error term, to N 2 − 2 / 2 l + 1 + ε when k = 3 and N k − 1 − 4 k − 2 / l + 1 k + l − 2 + ε when k ≥ 4 .

Fourier Optimization and Quadratic Forms

We prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer $\ell\ge 1$, we improve the error term in the partial sums of the number of representations of integers that are a multiple of $\ell$. This allows us to obtain unconditional Brun–Titchmarsh-type results in short intervals and a conditional Cramér-type result on the maximum gap between primes represented by a given positive definite quadratic form.

The number of representations of integers by generalized Bell ternary quadratic forms

For a positive definite ternary integral quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. A ternary quadratic form [Formula: see text] is said to be a generalized Bell ternary quadratic form if [Formula: see text] is isometric to [Formula: see text] for some nonnegative integers [Formula: see text]. In this paper, we give a closed formula for [Formula: see text] for a generalized Bell ternary quadratic form [Formula: see text] with [Formula: see text] and class number greater than [Formula: see text] by using the Minkowski–Siegel formula and bases for spaces of cusp forms of weight [Formula: see text] and level [Formula: see text] with [Formula: see text] consisting of eta-quotients.

Interesting identities involving weighted representations of integers as sums of arbitrarily many squares

We consider the number of ways to write an integer as a sum of squares, a problem with a long history going back at least to Fermat. The previous studies in this area generally fix the number of squares which may occur and then either use algebraic techniques or connect these to coefficients of certain complex analytic functions with many symmetries known as modular forms, from which one may use techniques in complex and real analysis to study these numbers. In this paper, we consider sums with arbitrarily many squares, but give a certain natural weighting to each representation. Although there are a very large number of such representations of each integer, we see that the weighting induces massive cancellation, and we furthermore prove that these weighted sums are again coefficients of modular forms, giving precise formulas for them in terms of sums of divisors of the integer being represented.

SHORT INTERVALS ASYMPTOTIC FORMULAE FOR BINARY PROBLEMS WITH PRIME POWERS, II

We improve some results in our paper [A. Languasco and A. Zaccagnini, ‘Short intervals asymptotic formulae for binary problems with prime powers’, J. Théor. Nombres Bordeaux30 (2018) 609–635] about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell _{1}}+p_{2}^{\ell _{2}}$ and $n=p^{\ell _{1}}+m^{\ell _{2}}$, where $\ell _{1},\ell _{2}\geq 2$ are fixed integers, $p,p_{1},p_{2}$ are prime numbers and $m$ is an integer. We also remark that the techniques here used let us prove that a suitable asymptotic formula for the average number of representations of integers $n=\sum _{i=1}^{s}p_{i}^{\ell }$, where $s$, $\ell$ are two integers such that $2\leq s\leq \ell -1$, $\ell \geq 3$ and $p_{i}$, $i=1,\ldots ,s$, are prime numbers, holds in short intervals.