Related papers: Sommes de G\'al et applications

In this paper, we present some asymptotic properties of the normalized inverse-Gaussian process. In particular, when the concentration parameter is large, we establish an analogue of the empirical functional central limit theorem, the…

We study the double character sum $\sum\limits_{\substack{m\leq X,\\m\odd}}\sum\limits_{\substack{n\leq Y,\\n\odd}}\leg mn$ and its smoothly weighted counterpart. An asymptotic formula with power saving error term was obtained by Conrey,…

We consider the sum $\sum 1/\gamma$, where $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0,T]$, and consider the behaviour of the sum as $T \to\infty$. We show that, after subtracting a…

Recently, Bordell\'{e}s, Dai, Heyman, Pan and Shparlinski in \cite{Igor} considered a partial sum involving the Euler totient function and the integer parts $\lfloor x/n\rfloor$ function. Among other things, they obtained reasonably tight…

In this paper, we investigate the conditional large values of the quadratic Dirichlet $L$-functions near the central point $s=1/2$. When $\sigma $ closes to $1/2$ within a suitable range, we show that $L(\sigma, \chi_d)$ have the…

Let $N(\sigma,T)$ denote the number of nontrivial zeros of the Riemann zeta function with real part greater than $\sigma$ and imaginary part between $0$ and $T$. We provide explicit upper bounds for $N(\sigma,T)$ commonly referred to as a…

A standard way to calculate the asymptotic behavior of integrals of the form \int_Wg(x)e^{-nh(x)}dx is the (continuous) Laplace asymptotic method. However, also discrete sums like \sum_{x\in W\cap\Lambda_n}g_n(x)e^{-nh_n(x)} have similar…

We study the average of the product of the central values of two $L$-functions of modular forms $f$ and $g$ twisted by Dirichlet characters to a large prime modulus $q$. As our principal tools, we use spectral theory to develop bounds on…

Let $N$ be a sufficiently large, odd integer. We prove an asymptotic formula for the number of representations of $N$ as the sum of three primes, one of which is smaller than a given $U$. By inserting the currently best zero-density…

In this paper, we develop Windschitl's approximation formula for the gamma function to two asymptotic expansions by using a little known power series. In particular, for $n\in \mathbb{N}$ with $n\geq 4$, we have \begin{equation*} \Gamma…

We estimate the number of integer solutions to decomposable form inequalities (both asymptotic estimates and upper bounds are provided) when the degree of the form and the number of variables are relatively prime. These estimates display…

In this paper we study the problem of prescribing Dirichlet eigenvalues on an arbitrary compact manifold $M$ of dimension $n\geq 3$ with a non-empty smooth boundary $\partial M$. We show that for any finite increasing sequence of real…

The sum of proper divisors function $s(n)$ has been studied for more than 2000 years. In this paper we study statistical properties of the related function $S_s(n) := \sum_{d \mid n} s(d)$. This function arises from a generalization of the…

Assuming the Riemann Hypothesis we establish an upper bound for the sum of the M{\" o}bius function up to $x$. Our method is based on estimating the frequency with which intervals of a given length can contain an unusual number of ordinates…

We establish asymptotic estimates for the least upper bounds of approximations in the uniform metric by Fourier sums of order $n-1$ of classes of $2\pi$-periodic Weyl--Nagy differentiable functions, $W^r_{\beta,p}, 1\le p\le \infty,…

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,…

In this paper we work out a Riemann-von Mangoldt type formula for the summatory function $\psi(x):=\sum_{g\in G, |g|\le x} \Lambda_{G}(g)$, where $G$ is an arithmetical semigroup (a Beurling generalized system of integers) and $\Lambda_{G}$…

We consider partial sums of a weighted Steinhaus random multiplicative function and view this as a model for the Riemann zeta function. We give a description of the tails and high moments of this object. Using these we determine the likely…

Assuming the Riemann Hypothesis we study negative moments of the Riemann zeta-function and obtain asymptotic formulas in certain ranges of the shift in $\zeta(s)$. For example, integrating $|\zeta(1/2+\alpha+it)|^{-2k}$ with respect to $t$…

Special values of the Lommel functions allow the calculation of Fresnel like integrals. These closed form expressions along with their asymptotic values are reported.