Related papers: Error estimates for Riemann sums of some singular …

This short note contains elementary evaluations of some Euler sums.

Estimates of some integrals related to variations of smooth functions are presented.

We develop a method for calculating Riemann sums using Fourier analysis.

In this paper, motivated by physical considerations, we introduce the notion of modified Riemann sums of Riemann-Stieltjes integrable functions, show that they converge, and compute them explicitely under various assumptions.

The approximation of integral functionals with respect to a stationary Markov process by a Riemann-sum estimator is studied. Stationarity and the functional calculus of the infinitesimal generator of the process are used to get a better…

We consider sums of the form $\sum \phi(\gamma)$, where $\phi$ is a given function, and $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such…

In this article we derive some polynomial inequalities for Mertens functions.

This paper evaluates some generalised Euler sums involving the digamma function.

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…

Error estimation is given for a regularized Shannon's sampling formulae, which was found to be accurate and robust for numerically solving partial differential equations.

This note contains some asymptotic formulas for the sums of various residue classes of Euler's phi-function.

We give the rate of convergence of some optimal lower Riemann-Stieltjes sums toward the integral.

This paper gives some results for the logarithm of the Riemann zeta-function and its iterated integrals. We obtain a certain explicit approximation formula for these functions. The formula has some applications, which are related with the…

A simple and elementary derivation of values at integer points for the Riemann's zeta and related functions is reported.

In this paper we give criteria about estimation of derivatives of the Riemann Zeta Function on the line $\sigma=1$.

An approximate formula for complex Riemann Xi function, previously developed, is used to refine Backlund's estimate of the number of zeros till a chosen imaginary coordinate

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…

We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis containing the M\"obius function. The estimate is remarkably sharp in comparison to estimates of other sums containing the M\"obius function.…

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…

An approach to constructing an upper bound for the Riemann-Farey sum is described.