Related papers: A fast algorithm to compute the Ramanujan-Deninger…

Let $G=(V,E)$ be an $n$-vertex graph, $L(G)\in \mathbb{R}^{n\times n}$ its Laplacian matrix, and let $\lambda_1(L(G))\ge \lambda_2(L(G))\ge \cdots\ge \lambda_n(L(G))=0$ denote its eigenvalues. For $1\le k\le n$, let $\varepsilon_k(G)=…

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…

We study positive solutions of the superlinear Lane-Emden inequality \(-\Delta u\ge \sigma u^q\), \(q>1\), on infinite locally finite weighted graphs and connected domains of such graphs. We first prove that solvability is equivalent to the…

We deal with negative square moments of Dirichlet $L$-functions. Summing over characters modulo $q$, we obtain an asymptotic formula for the negative second moment of $L(1,\chi)$ involving conductors. As an application, we give the improved…

This paper considers fast algorithms for operations on linearized polynomials. We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial…

We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $\varphi(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\leq q\leq 10$ and for $q=12,…

We consider the numerical solution of the equation - \Delta u - f(u) = g, for the unknown u satisfying Dirichlet conditions in a bounded domain. The nonlinearity f has bounded, continuous derivative. The algorithm uses the finite element…

The graph Fourier transform (GFT) is in general dense and requires O(n^2) time to compute and O(n^2) memory space to store. In this paper, we pursue our previous work on the approximate fast graph Fourier transform (FGFT). The FGFT is…

The purpose of this article is twofold. First, we introduce the constants $\zeta_k(\alpha,r,q)$ where $\alpha \in (0,1)$ and study them along the lines of work done on Euler constant in arithmetic progression $\gamma(r,q)$ by Briggs,…

For rational $\alpha$, the fractional partition functions $p_\alpha(n)$ are given by the coefficients of the generating function $(q;q)^\alpha_\infty$. When $\alpha=-1$, one obtains the usual partition function. Congruences of the form…

The generalized Stieltjes constants $\gamma\_n(v)$ are, up to a simple scaling factor, the Laurent series coefficients of the Hurwitz zeta function $\zeta(s,v)$ about its unique pole $s = 1$. In this work, we devise an efficient algorithm…

We define an absolutely convergent series for the upper incomplete Gamma function $\Gamma(s,z)$ for $z\geq 1$ and $s\in \mathbb{C}$. We express this series using certain polynomials which we define using the Stirling numbers of the first…

Let $X$ denote a connected $(q+1)$-regular undirected graph of finite order $n$. The graph $X$ is called Ramanujan whenever $$ |\lambda|\leq 2q^{\frac{1}{2}} $$ for all nontrivial eigenvalues $\lambda$ of $X$. We consider the variant…

The two-parametric Mittag-Leffler function (MLF), $E_{\alpha,\beta}$, is fundamental to the study and simulation of fractional differential and integral equations. However, these functions are computationally expensive and their numerical…

Dirichlet's theorem guarantees infinitely many primes in each reduced residue class modulo q, but the analytic mechanism underlying this separation is often difficult to visualize directly. In this article we construct simplified…

We express the Riemann zeta function $\zeta\left(s\right)$ of argument $s=\sigma+i\tau$ with imaginary part $\tau$ in terms of three absolutely convergent series. The resulting simple algorithm allows to compute, to arbitrary precision,…

Generalized Davenport-Schinzel sequences are sequences that avoid a forbidden subsequence and have a sparsity requirement on their letters. Upper bounds on the lengths of generalized Davenport-Schinzel sequences have been applied to a…

We exhibit an algorithm to compute a Dirichlet domain for a cofinite Fuchsian group Gamma. As a consequence, we compute the invariants of Gamma, including an explicit finite presentation for Gamma.

Let $f(n)$ be a multiplicative function with $|f(n)|\leq 1, q$ be a prime number and $a$ be an integer with $(a, q)=1, \chi$ be a non-principal Dirichlet character modulo $q$. Let $\varepsilon$ be a sufficiently small positive constant, $A$…

We give an new arithmetic algorithm to compute the generalized Discrete Fourier Transform (DFT) over finite groups $G$. The new algorithm uses $O(|G|^{\omega/2 + o(1)})$ operations to compute the generalized DFT over finite groups of Lie…