Related papers: An alternative to the Euler--Maclaurin formula: Ap…
We present highlights of computations of the Riemann zeta function around large values and high zeros. The main new ingredient in these computations is an implementation of the second author's fast algorithm for numerically evaluating…
For positive integers $p_1,p_2,\ldots,p_k,q$ with $q>1$, we define the Euler $T$-sum $T_{p_1p_2\cdots p_k,q}$ as the sum of those terms of the usual infinite series for the classical Euler sum $S_{p_1p_2\cdots p_k,q}$ with odd denominators.…
The Expectation Maximisation (EM) algorithm is widely used to optimise non-convex likelihood functions with latent variables. Many authors modified its simple design to fit more specific situations. For instance, the Expectation (E) step…
Dating back to Euler, in classical analysis and number theory, the Hurwitz zeta function $$ \zeta(z,q)=\sum_{n=0}^{\infty}\frac{1}{(n+q)^{z}}, $$ the Riemann zeta function $\zeta(z)$, the generalized Stieltjes constants $\gamma_k(q)$, the…
We propose a generic algorithm for computing the inverses of a multiplicative function under the assumption that the set of inverses is finite. More generally, our algorithm can compute certain functions of the inverses, such as their power…
Multiple q-zeta values are a 1-parameter generalization (in fact, a q-analog) of the multiple harmonic sums commonly referred to as multiple zeta values. These latter are obtained from the multiple q-zeta values in the limit as q tends to…
A sharper estimate for the summatory Euler phi function $\sum_{n \leq x} \varphi(n)$ is presented in this work. It improves the established estimate in the current mathematical literature. In addition, an estimate for its reciprocal…
We propose several variants of the Frank-Wolfe algorithm to minimize a sum of functions. The main proposed algorithm is inspired from the dual averaging scheme of Nesterov adapted for Frank Wolfe in a stochastic setting. A distributed…
An extension of the theory of the Iterated Logarithmic Algebra gives the logarithmic analog of a Sheffer or Appell sequence of polynomials. This leads to several examples including Stirling's formula and a logarithmic version of the…
Using Euler transformation of series we relate values of Hurwitz zeta function at integer and rational values of arguments to certain rapidly converging series where some generalized harmonic numbers appear. The form of these generalized…
While teaching a course on integral equations, I noticed that a straightforward combination of Neumann series and Fourier series for the resolvent (or the solution) of an integral equation has good approximation qualities. This short…
We define the generalized-Euler-constant function $\gamma(z)=\sum_{n=1}^{\infty} z^{n-1} (\frac{1}{n}-\log \frac{n+1}{n})$ when $|z|\leq 1$. Its values include both Euler's constant $\gamma=\gamma(1)$ and the "alternating Euler constant"…
We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \le x$ for which the Alladi-Erd\H{o}s function $A(n) =…
We give explicit expressions (or at least an algorithm of obtaining such expressions) of the coefficients of the Laurent series expansions of the Euler-Zagier multiple zeta-functions at any integer points. The main tools are the…
We equip the space of lattice cones with a coproduct which makes it a connected cograded colagebra. The exponential sum and exponential integral on lattice cones can be viewed as linear maps on this space with values in the space of…
Recently, Dil and Boyadzhiev \cite{AD2015} proved an explicit formula for the sum of multiple harmonic numbers whose indices are the sequence $\left( {{{\left\{ 0 \right\}}_r},1} \right)$. In this paper we show that the sums of multiple…
Translation of "Methodus succincta summas serierum infinitarum per formulas differentiales investigandi" (1780). Euler wants to represent some given series of functions S(x)=X(x)+X(x+1)+X(x+2)+etc. in a different way. He writes S as a…
This study presents explicit evaluations of the series \begin{equation*} \sum_{k=1}^\infty \frac{H_{k/n}^{(p)}}{k^q} \quad \text{and} \quad \sum_{k=1}^\infty \frac{(-1)^k H_{k/2n}^{(p)}}{k^q}, \quad p,q,n \in \mathbb{Z}_{\ge 1},\; q \ne 1,…
A recently developed analytical method for systematic improvement of the convergence of path integrals is used to derive a generalization of Euler's summation formula for path integrals. The first $p$ terms in this formula improve…
The hyperharmonic numbers h_{n}^{(r)} are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers: {\sigma}(r,m)=\sum_{n=1}^{\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in terms of…