Alexey Kuznetsov

Alexey Kuznetsov

The function $Q(x):=\sum_{n\ge 1} (1/n) \sin(x/n)$ was introduced by Hardy and Littlewood [10] in their study of Lambert summability, and since then it has attracted attention of many researchers. In particular, this function has made a surprising appearance in the recent disproof by Alzer, Berg and Koumandos [1] of a conjecture by Clark and Ismail [2]. More precisely, Alzer et. al. [1] have shown that the Clark and Ismail conjecture is true if and only if $Q(x)\ge -π/2$ for all $x>0$. It is known that $Q(x)$ is unbounded in the domain $x \in (0,\infty)$ from above and below, which disproves the Clark and Ismail conjecture, and at the same time raises a natural question of whether we can exhibit at least one point $x$ for which $Q(x) < -π/2$. This turns out to be a surprisingly hard problem, which leads to an interesting and non-trivial question of how to approximate $Q(x)$ for very large values of $x$. In this paper we continue the work started by Gautschi in [7] and develop several approximations to $Q(x)$ for large values of $x$. We use these approximations to find an explicit value of $x$ for which $Q(x)<-π/2$.

Alexey Kuznetsov

We develop approximations for the Riemann zeta function that enable high-precision computation within the critical strip and other vertical strips. These approximations combine the main sum of the Riemann-Siegel formula with a simple approximation of the remainder term, which involves only elementary functions and certain precomputed coefficients obtained via Gaussian quadrature. Additionally, we provide approximations for the derivative of the Riemann zeta function and present extensive numerical evidence demonstrating the accuracy of these approximations.

Alexey Kuznetsov, Armin Mohammadioroojeh

We present a new method for approximating real-valued functions on ${\mathbb R}^+$ by linear combinations of exponential functions with complex coefficients. The approach is based on a multi-point Padé approximation of the Laplace transform and employs a highly efficient continued fraction technique to construct the corresponding rational approximant. We demonstrate the accuracy of this method through a variety of examples, including the Gaussian function, probability density functions of the lognormal and Gompertz-Makeham distributions, the hockey stick and unit step functions, as well as a function arising in the approximation of the gamma and Barnes $G$-functions.