Related papers: Omega theorem for fractional sigma function

We introduce Omega functions that generalize Euler Gamma functions and study the functional difference equation they satisfy. Under a natural exponential growth condition, the vector space of meromorphic solutions of the functional equation…

The study of graph discrepancy problems, initiated by Erd\H{o}s in the 1960s, has received renewed attention in recent years. In general, given a $2$-edge-coloured graph $G$, one is interested in embedding a copy of a graph $H$ in $G$ with…

We consider the numerical solution of the fractional Laplacian of index $s\in(1/2,1)$ in a bounded domain $\Omega$ with homogeneous boundary conditions. Its solution a priori belongs to the fractional order Sobolev space ${\widetilde…

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…

In this article, we investigate the convergence rate of the discrete-time Clark--Ocone formula provided by Akahori--Amaba--Okuma [1]. In that paper, they mainly focus on the $L_{2}$-convergence rate of the first-order error estimate related…

Let $\phi(n)$ be the Euler totient function and $\sigma(n)$ denote the sum of divisors of $n$. In this note, we obtain explicit upper bounds on the number of positive integers $n\leq x$ such that $\phi(\sigma(n)) > cn$ for any $c>0$. This…

Following (Kolpakov et al., 2013; Gawrychowski and Manea, 2015), we continue the study of {\em $\alpha$-gapped repeats} in strings, defined as factors $uvu$ with $|uv|\leq \alpha |u|$. Our main result is the $O(\alpha n)$ bound on the…

We investigate the distribution of the Riemann zeta-function on the line $\Re(s)=\sigma$. For $\tfrac 12 < \sigma \le 1$ we obtain an upper bound on the discrepancy between the distribution of $\zeta(s)$ and that of its random model,…

Estimating the second frequency moment of a stream up to $(1\pm\varepsilon)$ multiplicative error requires at most $O(\log n / \varepsilon^2)$ bits of space, due to a seminal result of Alon, Matias, and Szegedy. It is also known that at…

Assuming the Generalized Riemann Hypothesis (GRH), we utilize the long resonator method to derive $\Omega$-results for the family of quadratic Dirichlet $L$-functions $L(\sigma, \chi_d)$, where $d$ runs over all fundamental discriminants…

We calculate some infinite sums containing the digamma function in closed-form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as…

The purpose of this article is to establish new lower bounds for the sums of powers of eigenvalues of the Dirichlet fractional Laplacian operator $(-\Delta)^{\alpha/2}|_{\Omega}$ restricted to a bounded domain $\Omega\subset{\mathbb R}^d$…

We introduce an algorithm to compute the functions belonging to a suitable set ${\mathscr F}$ defined as follows: $f\in {\mathscr F}$ means that $f(s,x)$, $s\in A\subset {\mathbb R}$ being fixed and $x>0$, has a power series expansion…

We introduce approximation functions of $li(x)$ for all $x\ge e$: (1) $\displaystyle li_{\underline{\omega},\alpha}(x) = \frac{x}{\log(x)}\left( \alpha\frac{\underline{m}!}{\log^{\underline{m}}(x)} +…

We consider some properties of integrals considered by Hardy and Koshliakov, and which have also been further extended recently by Dixit. We establish a new general integral formula from some observations about the digamma function. We also…

We present a class of multiplicative functions $f:\mathbb{N}\to\mathbb{C}$ with bounded partial sums. The novelty here is that our functions do not need to have modulus bounded by $1$. The key feature is that they pretend to be the constant…

For $V\sim \alpha \log\log T$ with $0<\alpha<2$, we prove \[ \frac{1}{T}\text{meas}\{t\in [T,2T]: \log|\zeta(1/2+ {\rm i} t)|>V\}\ll \frac{1}{\sqrt{\log\log T}} e^{-V^2/\log\log T}. \] This improves prior results of Soundararajan and of…

In this article, we investigate large values of Dirichlet character sums with multiplicative coefficients $\sum_{n\le N}f(n)\chi(n)$. We prove an Omega result in the region $\exp((\log q)^{\frac12+\varepsilon})\le N\le\sqrt q$, where $q$ is…

If $d$ is not a perfect square, we define $T(d)$ as the length of the minimal period of the simple continued fraction expansion for $\sqrt{d}$. Otherwise, we put $T(d) = 0$. In the recent paper (2024), F.Battistoni, L.Greni\'{e} and…

In previous work, the author gave upper bounds for the shifted moments of the zeta function \[ M_{{\alpha},{\beta}}(T) = \int_T^{2T} \prod_{k = 1}^m |\zeta(\tfrac{1}{2} + i (t + \alpha_k))|^{2 \beta_k} dt \] introduced by Chandee, where…