Related papers: Estimates for Character Sums with Various Convolut…

Consider the Weil sum $W_{F,d}(u)=\sum_{x \in F} \psi(x^d+u x)$, where $F$ is a finite field of characteristic $p$, $\psi$ is the canonical additive character of $F$, $d$ is coprime to $|F^*|$, and $u \in F^*$. We say that $W_{F,d}(u)$ is…

We show in a quantitative way that any odd character $\chi$ modulo $q$ of fixed order $g \geq 2$ satisfies the property that if the P\'{o}lya-Vinogradov inequality for $\chi$ can be improved to $$\max_{1 \leq t \leq q} \left|\sum_{n \leq t}…

Let $p$ be an odd prime. In the paper we collect the author's various conjectures on congruences modulo $p$ or $p^2$, which are concerned with sums of binomial coefficients, Lucas sequences, power residues and special binary quadratic…

In this paper, we present the character analogue of the Boole summation formula. Using this formula, an integral representation is derived for the alternating Dirichlet $L-$function and its derivative is evaluated at $s=0$. Some…

In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are…

Let f(z) = sum_n a(n) n^{(k-1)/2} e(nz) be a cusp form for Gamma_0(N), character chi and weight k geq 4. Let q(x) = x^2 + sx + t be a polynomial with integral coefficients. It is shown that sum_{n \leq X} a(q(n)) = cX + O(X^{6/7+eps}) for…

Let $\mathbb{F}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a non-empty subset of $\mathbb{F}_p.$ In this paper we give an explicit version of the sum-product estimate of Bourgain, Katz, Tao and Bourgain,…

First we reprove two results in additive number theory due to Dombi and Chen & Wang, respectively, on the number of representations of n as the sum of two odious or evil numbers, using techniques from automata theory and logic. We also use…

We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solution in boxes with the side…

Let $R$ be a finite ring with unity, $\psi: R \to \mathbb{C}^\times$ be an additive character of $R$, and \( \chi_0 \) be the principal multiplicative character ($i.e.$, $\chi_0(x) = 1 \quad \text{for all } x \in R^\times$), then the Gauss…

We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of…

The explicit formulas for the sums of positive powers of the integers $s_i$ unrepresentable by the triple of integers $d_1,d_2,d_3\in {\mathbb N}, \gcd(d_1,d_2,d_3)=1$, are derived.

Let $p$ be a prime. For $p=2$, the fields of values of the complex irreducible characters of finite groups whose degrees are not divisible by $p$ have been classified; for odd primes $p$, a conjectural classification has been proposed. In…

For a prime p and base b, the collision invariant $S_{\ell}(p)$, introduced in the companion paper, is a function of $p \bmod b^{\ell+1}$ and therefore lives on the finite group $(\mathbb{Z}/b^{\ell+1}\mathbb{Z})^{\times}$. Its Fourier…

We review the calculations of form factors and coupling constants in vertices with charm mesons in the framework of QCD sum rules. We first discuss the motivation for this work, describing possible applications of these form factors to…

We compute the QCD form factor resumming threshold logarithms in B --> X_c + l + nu_l decays to next-to-leading logarithmic approximation. We present an interpolation formula including soft as well as collinear effects softened by the…

Ingham studied two types of convolution sums of the divisor function, namely the shifted convolution sum $\sum_{n \le N} d(n) d(n+h)$ and the additive convolution sum $\sum_{n < N} d(n) d(N-n)$ for integers $N, h$ and derived their…

Let $q\ge3$ be an integer, $\chi$ be a Dirichlet character modulo $q$, and $L(s,\chi)$ denote the Dirichlet $L$-functions corresponding to $\chi$. In this paper, we show some special function series, and give some new identities for the…

In this paper, we will prove the non-trivial bound for the weighted average version of shifted convolution sum for $GL(3)\times GL(2)$, i.e. for any $\epsilon >0$ and $X^{1/4+\delta} \leq H \leq X$ with $\delta >0$, \[…

We study the sum of the squares of the irreducible character degrees not divisible by some prime $p$, and its relationship with the the corresponding quantity in a $p$-Sylow normalizer. This leads to study a recent conjecture by E.…