Related papers: Three-dimensional exponential sums under constant …

We examine a family of three-dimensional exponential sums with monomials and provide estimates which are in some instances sharper than those stemming from approaches entailing the use of existing bounds pertaining to analogous sums.

We study the triple convolution sum of the generalised divisor functions $$\sum_{n\leq x} d_k(n+h)d_l(n)d_m(n-h),$$ where $h \le x^{1-\epsilon}$ for any $\epsilon>0$ and $d_k(n)$ denotes the generalised divisor function which counts the…

By using various expansions of the parametric digamma function and the method of residue computations, we study three variants of the linear Euler sums, related Hoffman's double $t$-values and Kaneko-Tsumura's double $T$-values, and…

This work presents a study of perturbations of symmetric Boolean functions. In particular, it establishes a connection between exponential sums of these perturbations and Diophantine equations of the form $$ \sum_{l=0}^n \binom{n}{l}…

We show that the expected asymptotic for the sums $\sum_{X < n \leq 2X} \Lambda(n) \Lambda(n+h)$, $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$, and $\sum_{X < n \leq 2X} \Lambda(n) d_k(n+h)$ hold for almost all $h \in [-H,H]$, provided that…

We examine convergent representations for the sum of a decaying exponential and a Bessel function in the form \[\sum_{n=1}^\infty \frac{e^{-an}}{(\frac{1}{2} bn)^\nu}\,J_\nu(bn),\] where $J_\nu(x)$ is the Bessel function of the first kind…

In this paper, we define a parametric variant of generalized Euler sums and call them the (alternating) parametric Euler $T$-sums. By using the contour integration method and residue theorem, we establish several explicit formulae for the…

We examine exponential sums of the form $\sum_{n \le X} w(n) e^{2\pi i\alpha n^k}$, for $k=1,2$, where $\alpha$ satisfies a generalized Diophantine approximation and where $w$ are different arithmetic functions that might be multiplicative,…

We consider a three-dimensional Fourier integral in which the exponent in the exponential factor is the product of some phase function and a large parameter. The asymptotics of this integral is sought when the large parameter tends to…

We produce nontrivial asymptotic estimates for shifted sums of the form $\sum a(h)b(m)c(2m-h)$, in which $a(n),b(n),c(n)$ are un-normalized Fourier coefficients of holomorphic cusp forms. These results are unconditional, but we demonstrate…

We study the mean value of sixth power of some exponential sums following an idea due to Bombieri and Iwaniec. The result applies to Weyl's inequality, following an idea due to Heath-Brown

In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for…

We establish bounds of triple exponential sums with mixed exponential and linear function. The method we use is by Shparlinski together with a bound of additive energy from Roche-Newton, Rudnev and Shkredov.

A sum of observations derived by a simple random sampling design from a population of independent random variables is studied. A procedure finding a general term of Edgeworth asymptotic expansion is presented. The Lindeberg condition of…

An improved $\eps$ expansion in the $d$-dimensional ($d > 2$) stochastic theory of turbulence is constructed by taking into account pole singularities at $d \to 2$ in coefficients of the $\eps$ expansion of universal quantities.…

We study three classes of combinatorial sums involving central binomial coefficients and harmonic numbers, odd harmonic numbers, and even indexed harmonic numbers, respectively. In each case we use summation by parts to derive recursive…

The universal perturbative invariants of rational homology spheres can be extracted from the Chern-Simons partition function by combining perturbative and nonperturbative results. We spell out the general procedure to compute these…

We consider a multiple arithmetical sum involving the Moebius function which despite its elementary appearance is in fact of a highly intriguing nature. We establish an asymptotic formula for the quadruple case that raises the first…

We establish a new bound for the exponential sum \begin{eqnarray*} \sum_{x\in\mathcal{X}}\Big|\sum_{y\in \mathcal{Y}}\gamma(y)\exp(2\pi i a \lambda^{xy}/p)\Big|, \end{eqnarray*} where $\lambda$ is an element of the residue ring modulo a…

Recently, Li obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture. As a consequence, he showed the positivity of this sum. His result was based on a sieving…