Related papers: Mean square of the error term in the asymmetric ma…

We prove uniform versions of two classical results in analytic number theory. The first is an asymptotic for the number of points of a complete lattice $\Lambda \subseteq \mathbb{R}^d$ inside the $d$-sphere of radius $R$. In contrast to…

Asymmetric systematic errors arise when there is a non-linear dependence of a result on a nuisance parameter. Their combination is traditionally done by adding positive and negative deviations separately in quadrature. There is no sound…

We give a general statement of the convolution method so that one can provide explicit asymptotic estimations for all averages of square-free supported arithmetic functions that have a sufficiently regular order on the prime numbers and…

In this paper, we find asymptotic formula for the following sum with explicit error term: \[M_{x}(g_{1}, g_{2}, g_3)=\frac{1}{x}\sum_{n\le x}g_{1}(F_1(n))g_{2}(F_2(n))g_{3} (F_3(n)),\] where $F_1(x), F_2(x)$ and $F_3(x)$ are polynomials…

Let $\beta$ be a positive integer. A generalization of the Ramanujan sum due to Cohen is given by \begin{align} c_{q,\beta }(n) := \sum\limits_{{{(h,{q^\beta })}_\beta } = 1} {{e^{2\pi inh/{q^\beta }}}}, \nonumber \end{align} where $h$…

Friedlander and Iwaniec investigated integral solutions $(x_1,x_2,x_3)$ of the equation $x_1^2+x_2^2-x_3^2=D$, where $D$ is square-free and satisfies the congruence condition $D\equiv 5\bmod{8}$. They obtained an asymptotic formula for…

A celebrated result of Hal\'asz describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions…

Let $k$ be a number field. In this paper, we give a formula for the mean value of the square of class numbers times regulators for certain families of quadratic extensions of $k$ characterized by finitely many local conditions. We approach…

We evaluate the smoothed first moment of central values of a family of qudratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series. The asymptotic formula we obtain has an error term of size…

We derive sufficient conditions for asymptotic and monotone exponential decay in mean square of solutions of the geometric Brownian motion with delay. The conditions are written in terms of the parameters and are explicit for the case of…

Let $f(n)$ be an arithmetic function with $f(n) \ll n^\alpha$ for some $\alpha\in[0,1)$ and let $\lfloor .\rfloor $ denote the integer part function. In this paper, we evaluate asymptotically the sums $$\sum_{n_{1}n_{2}\leq x}f \left(…

We examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by \[S_{\nu,p}(a)=\sum_{n\geq 1} (an^p/2)^{-\nu} K_\nu(an^p)\qquad (a>0,\ 0\leq\nu<1)\] as the parameter $a\to 0+$, where $p$…

We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length $T^{\frac14 - \varepsilon}$. Our work relies crucially on Watt's theorem on averages of Kloosterman fractions. In the…

In this article we study the arithmetic mean value $\Sigma(n)$ of the square roots of the first $n$ integers. For this quantity, we develop an asymptotic expression, and derive a formula for its integer part which has been conjectured…

We give a relatively simple proof that \[ \int _0^1\left |\sum _{n\leq x}d(n)e(n\alpha )\right |d\alpha \asymp \sqrt x.\]

We prove new exact formulas for the generalized sum-of-divisors functions, $\sigma_{\alpha}(x) := \sum_{d|x} d^{\alpha}$. The formulas for $\sigma_{\alpha}(x)$ when $\alpha \in \mathbb{C}$ is fixed and $x \geq 1$ involves a finite sum over…

In this paper, we derive the following asymptotic formula $$ \mathop{{\sum}'}_{n\leqslant x}\dfrac{r(n)}{r(n+1)} = {x}{(\ln x)^{-3/4}}(c+o(1)),\ \ x \to +\infty,$$ where $r(n)$ is the number of representations of $n$ as a sum of two…

The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for $X^{\frac{2}{3}+\epsilon} < H <X^{1-\epsilon},$ there are constants $B_{h}$ such that $$ \sum_{X\leq n \leq 2X}…

Let $F$ be a non-degenerate quadratic form on an $n$-dimensional vector space $V$ over the rational numbers. One is interested in counting the number of zeros of the quadratic form whose coordinates are restricted in a smoothed box of size…

Given $n$ positive integers $a_1,a_2,\dots,a_n$, and a positive integer right hand side $\beta$, we consider the feasibility version of the subset sum problem which is the problem of determining whether a subset of $a_1,a_2,\dots,a_n$ adds…