In this paper we present a new formula relating Stieltjes numbers γn and Laurent coefficinets ηn of logarithmic derivative of the Riemann’s zeta function. Using it we derive an explicit formula for the oscillating part of Li’s numbers ∼ λn which are connected with the Riemann hypothesis.

In this paper we provide sharp error bounds in approximating the weighted Riemann-Stieltjes integral R b a f (t) g (t) d (t) by the weighted trapezoidal rule f(a)+f(b) 2 R b a g (t) d (t) : Applications for continuous functions of selfadjoint operators in complex Hilbert spaces are given as well.

We present analytic properties and extensions of the constants ck appearing in the Báez-Duarte criterion for the Riemann hypothesis. These constants are the coefficients of Pochhammer polynomials in a series representation of the reciprocal of the Riemann zeta function. We present generalizations of this representation to the Hurwitz zeta and many other special functions. We relate the correspo...

We obtain existence and uniqueness results for the solutions of a system Caputo fractional differential equations which contain sequential derivatives, integral terms, two positive parameters, supplemented with general coupled Riemann–Stieltjes boundary conditions. The proofs our are based on Banach fixed point theorem Leray–Schauder alternative.

We use Cavalieri’s principle to develop a novel integration technique which we call Cavalieri integration. Cavalieri integrals differ from Riemann integrals in that non-rectangular integration strips are used. In this way we can use single Cavalieri integrals to find the areas of some interesting regions for which it is difficult to construct single Riemann integrals. We also present two method...

In this paper, the existence of absolutely continuous solutions and some properties will be studied for a nonlocal boundary value problem state-dependent differential equation. The infinite-point condition Riemann–Stieltjes integral also considered. Some examples provided to illustrate our results.

Aturan titik tengah double dimodifikasi untuk mengaproksimasi integral RiemannStieltjes. Koefisien dari metode ini didapat dengan menyelesaikannya sistem persamaan nonlinear yang penerapan fungsi monomial sampai pangkat tertentu. Selanjutnya dibahas bentuk error diusulkan. Pada bagian akhir diberikan hasil komputasi numerik diusulkan dan dibandingkan metode-metode sebelumnya.