Related papers: Matrix Kloosterman sums modulo prime powers

We consider the periods of the linear congruential and the power generators modulo $n$ and, for fixed choices of initial parameters, give lower bounds that hold for ``most'' $n$ when $n$ ranges over three different sets: the set of primes,…

Recent work has introduced the study of graphical properties of cyclic supercharacters, functions $\mathbb{Z}/n\mathbb{Z}\to \mathbb{C}$ whose values are exponential sums with close connections to Gauss sums and Gaussian periods. Plots of…

A generalization of recent group-theoretic matrix multiplication algorithms to an analogue of the theory of partial matrix multiplication is presented. We demonstrate that the added flexibility of this approach can in some cases improve…

Poincar\'e in 1911 and Petersson in 1932 gave the now classical expression for the parabolic Fourier coefficients of holomorphic Poincar\'e series in terms of Bessel functions and Kloosterman sums. Later, in 1941, Petersson introduced…

We obtain several asymptotic formulas for the sum of the divisor function $\tau(n)$ with $n \le x$ in an arithmetic progressions $n \equiv a \pmod q$ on average over $a$ from a set of several consecutive elements from set of reduced…

Let p be a prime. The Hilbert-Kunz multiplicity, mu, of the element sum(x_i^(d_i)) of (Z/p)[x_1,..., x_s] depends on p in a complicated way. We calculate the limit of mu as p -> infinity. In particular when each d_i is 2 we show that the…

This work brings techniques from the theory of recurrent integer sequences to the problem of balancedness of symmetric Boolean functions. In particular, the periodicity modulo $p$ ($p$ odd prime) of exponential sums of symmetric Boolean…

We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on…

We prove, for a sufficiently small subset $\mathcal{A}$ of a prime residue field, an estimate on the number of solutions to the equation $(a_1-a_2)(a_3-a_4) = (a_5-a_6)(a_7-a_8)$ with all variables in $\mathcal{A}$. We then derive new…

The Cram\'er-Granville conjecture is an upper bound on prime gaps, $g_n = p_{n+1} - p_n < \cCramer \, \log^2 p_n$ for some constant $\cCramer \geq 1$. Using a formula of Selberg, we first prove the weaker summed version: $\sum_{n=1}^N g_n <…

In the proposed matrix primes, through which one can readily generate a sequence of primes. The paper also proposes a number of theorems proved by which an infinite number of prime numbers twins

In this paper, we construct eight infinite families of ternary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the special orthogonal group $SO^{-}(2n,q)$. Here ${q}$ is a power of three.…

We obtain an upper bound for the distribution of primes in the form $n^4 + k$ up to $x$, averaged over $k$ with small square-full part. As a corollary, we show that for almost all $k$, there is an expected amount of primes in the form $n^4…

Sums over inverse s-th powers of semiprimes and k-almost primes are transformed into sums over products of powers of ordinary prime zeta functions. Multinomial coefficients known from the cycle decomposition of permutation groups play the…

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…

We prove lower bounds of order $n\log n$ for both the problem to multiply polynomials of degree $n$, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower…

Holomorphic almost modular forms are holomorphic functions of the complex upper half plane which can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in $\SL(2,\ZZ)$. It is…

In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we…

In this work, new closed-form formulas for the matrix exponential are provided. Our method is direct and elementary, it gives tractable and manageable formulas not current in the extensive literature on this essential subject. Moreover,…

We consider the initial boundary value problem for a model system of one-dimensional equations which describe unsteady polytropic motions of a mixture of viscous compressible fluids. We prove the global existence and uniqueness theorem for…