Related papers: Exponential Sums and Congruences with Factorials

In this paper, we pose lots of challenging conjectures on congruences for the sums involving binomial coefficients and Ap\'ery-like numbers modulo $p^3$, where $p$ is an odd prime.

The main object of this paper is to find closed form expressions for finite and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the number of distinct prime factors of $n$. We then derive general convergence criteria…

Let $P_{r}$ denote an integer with at most $r$ prime factors counted with multiplicity. In this paper we prove that for some $\lambda < \frac{1}{12}$, the inequality $\{\sqrt{p}\}<p^{-\lambda}$ has infinitely many solutions in primes $p$…

Let $p_{-k}(n)$ enumerate the number of $k$-colored partitions of $n$. In this paper, we establish some infinite families of congruences modulo 25 for $k$-colored partitions. Furthermore, we prove some infinite families of Ramanujan-type…

In this short note, we closely follow the approach of Green and Tao to extend the best known bound for recurrence modulo 1 from squares to the largest possible class of polynomials. The paper concludes with a brief discussion of a…

We study the distribution of consecutive sums of two squares in arithmetic progressions. If $\{E_n\}_{n \in \mathbb{N}}$ is the sequence of sums of two squares in increasing order, we show that for any modulus $q$ and any congruence classes…

Suppose that some harmonic analysis arguments have been invoked to show that the indicator function of a set of residue classes modulo some integer has a large Fourier coefficient. To get information about the structure of the set of…

We show that if an exponential sum with multiplicative coefficients is large then the associated multiplicative function is "pretentious". This leads to applications in the circle method, and a natural interpretation of the local-global…

Given a natural number $n \geq 4$ we show that there exists infinitely many polynomials $f_{n}(x):= \prod_{i=1}^{n} (x^{2} - a_{i})$ such that (i) $f_{n}(x)$ has a root modulo every positive integer, (ii) $f_{n}(x)$ has no rational roots,…

Let f be a polinomial with coefficients in a finite field F. Let $\Psi : F \to C^{\ast}$ be a non-trivial additive character. In this paper we give bounds for the exponential sums $\sum_{x\in F^n} \Psi (Tr_{F/F_p} (f(x)))$ in some cases…

For a prime number $p$ and integer $x$ with $\gcd(x,p)=1$ let $\overline{x}$ denote the multiplicative inverse of $x$ modulo $p.$ In the present paper we are interested in the problem of distribution modulo $p$ of the sequence $$…

We discuss the asymptotic expansions of certain products of Bernoulli numbers and factorials, e.g., \[ \prod_{\nu=1}^n |B_{2\nu}| \quad \text{and} \quad \prod_{\nu=1}^n (k \nu)!^{\nu^r} \quad \text{as} \quad n \to \infty \] for integers $k…

Let $p$ be an odd prime and let $f(x)=\sum_{i=1}^ka_ix^{p^{\alpha_i}+1}\in\Bbb F_{p^n}[x]$, where $0\le \alpha_1<...<\alpha_k$. We consider the exponential sum $S(f,n)=\sum_{x\in\Bbb F_{p^n}}e_n(f(x))$, where $e_n(y)=e^{2\pi…

Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove…

We find a new approach to computing the remainder of a polynomial modulo $x^n-1$; such a computation is called modular enumeration. Given a polynomial with coefficients from a commutative $\mathbb{Q}$-algebra, our first main result…

Let $C_n=n2^n+1$ denote the $n$th Cullen number. There has been recent interest in finding all Cullen numbers having a given Diophantine property. We prove that, for a fixed integer $k$ and bounded integers $a_1,\ldots,a_k$, the greatest…

Let $P$ be a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions $n\leq N$ to $n! = P(x)$ which yields a power saving over the trivial bound. In particular, this applies…

We study averages over squarefree moduli of the size of exponential sums with polynomial phases. We prove upper bounds on various moments of such sums, and obtain evidence of un-correlation of exponential sums associated to different…

A concise and elementary derivation of the complete asymptotic expansion for the factorial function $n!$ is presented. This treatment produces a new expression for the coefficients, and it brings to light the simple relationship between the…

It is proven that, in any given base, there are infinitely many palindromic numbers having at most six prime divisors, each relatively large. The work involves equidistribution estimates for the palindromes in residue classes to large…