Related papers: On certain sums over primes and the Riesz function
The Euler-Maclaurin formula which relates a discrete sum with an integral, is generalised to the setting of Riemann-Stieltjes sums and integrals on stochastic processes whose paths are a.s. rectifiable, namely, continuous and with bounded…
In this work we prove analogues of Bessel inequality and Riesz-Fisher theorem in Hilbert spaces with respect to sequences. We apply our generalized Bessel inequality to the Hilbert spaces associated with the Normal, Beta, Gamma and certain…
We present new relationships between the work of H. Davenport and A. I. Popov. A new general formula involving the Von Mangoldt function is presented, as well as a criteria for the Riemann Hypothesis.
Various product and sum relationships are established using special functions, specifically involving Special functions. These relationships are derived from formulas inspired by the finite sum that incorporates the Hurwitz-Lerch zeta…
We exploit transformations relating generalized $q$-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as $\pi$, and to connect sums…
In this mainly expository article, we revisit some formal aspects of B{\'a}ez-Duarte's criterion for the Riemann hypothesis. In particular, starting from Weingartner's formulation of the criterion, we define an arithmetical function $\nu$,…
Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an explicit formula for the number of representations of an integer as a sum of $k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic factors…
The present book gives a systematic overview of function theory and the theory of Stieltjes integral. In particular, we give a detailed account of the theory of functions of bounded variation and of the theory of regulated functions (=…
This note is concerned with series of the forms $\sum f(a^n)$ and $\sum f(n^{-a})$ where f(a) possesses a Mellin transform and $a > 1$ or $a<0$ respectively. Integral representations are derived and used to transform these series in several…
We study arithmetic functions $\Phi(x;d,a)$, called prime running functions, whose value at $x$ sums the gaps between primes $p_k \equiv a\ (\text{mod}\ d)$ below $x$ and the next following prime $p_{k+1}$, up to $x$. (The following prime…
Recently, it was conjectured that the first generalized Stieltjes constant at rational argument may be always expressed by means of Euler's constant, the first Stieltjes constant, the $\Gamma$-function at rational argument(s) and some…
Approximative properties of linear summation methods of Fourier series are considered in the Orlicz type spaces ${\mathcal S}_{M}$. In particular, in terms of approximations by such methods, constructive characteristics are obtained for…
We develop a method for calculating Riemann sums using Fourier analysis.
In our previous work we found sufficient conditions to be imposed on the parameters of the generalized hypergeometric function in order that it be completely monotonic or of Stieltjes class. In this paper we collect a number of consequences…
In this short note, we obtain error estimates for Riemann sums of some singular functions.
Explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized.
By using Beta Dirichlet series and then Eisenstein series we ca represent primes with first a good approximation and an exact expression. This can be done with arbitrary prime (up to 10^101).
We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within…
Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases, we show that we obtain the best possible constant or that our bounds are tight in certain limits. We…
Let the summatory function of the M\"{o}bius function be denoted $M(x)$. We deduce in this article conditional results concerning $M(x)$ assuming the Riemann Hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the…