Related papers: The Riemann hypothesis - an elementary analytic ap…
By using the generalized Bernoulli numbers, we deduce new integral representations for the Riemann zeta function at positive odd-integer arguments. The explicit expressions enable us to obtain criteria for the dimension of the vector space…
The theory of Mellin transform is an incredibly useful tool in evaluating some of the well known results for the zeta function. Ramanujan in his quarterly reports \cite{1} gave a theorem for Mellin transform which is now known as…
We explore the class of probability distributions on the real line whose Laplace transform admits a strong upper bound of subgaussian type. Using Hadamard's factorization theorem, we extend the class $\mathfrak L$ of Newman and propose new…
We transform the counting function for the Riemann zeros into a Korringa-Kohn-Rostoker (KKR) determinant, assisted by Krein's theorem. This is based on our observation that the function derived from a few methods can all be recast into two…
In this paper, we present a proof of the Riemann hypothesis. We show that zeros of the Riemann zeta function should be on the line with the real value 1/2, in the region where the real part of complex variable is between 0 and 1.
We establish existence and pointwise estimates of fundamental solutions and Green's matrices for divergence form, second order strongly elliptic systems in a domain $\Omega \subseteq \mathbb{R}^n$, $n \geq 3$, under the assumption that…
We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating…
We prove Riemann hypothesis. Method is to show the convexity of function which has zeros on open critical strip the same as zeta function.
Assuming the generalized Riemann hypothesis, we prove quantitative estimates for the number of simple zeros on the critical line for the L-functions attached to classical holomorphic newforms.
In this work we perform a Green's function analysis of giant-dipole systems. First we derive the Green's functions of different magnetically field-dressed systems, in particular of electronically highly excited atomic species in crossed…
The conserved densities of hydrodynamic type system in Riemann invariants satisfy a system of linear second order partial differential equations. For linear systems of this type Darboux introduced Laplace transformations, generalising the…
Formulas for calculating the Riesz function, introduced by Marcel Riesz in connection with the Riemann hypothesis, are derived; and the behavior of the Riesz function is discussed.
The Christoffel problem is equivalent to the existence of convex solutions to the Laplace equation on the unit sphere $S^n$. Necessary and sufficient conditions have been found by Firey and Berg, using the Green function of the Laplacian on…
An integral formula is developed which applies to an essentially arbitrary function. An application is made to the Riemann zeta function.
We describe a numerical algorithm for evaluating the numbers of roots minus the number of poles contained in a region based on the argument principle with the function of interest being written as a Mellin transformation of a usually…
The exponential decay of lattice Green functions is one of the main technical ingredients of the Ba{\l}aban's approach to renormalization. We give here a self-contained proof, whose various ingredients were scattered in the literature. The…
The GUE Hypothesis, which concerns the distribution of zeros of the Riemann zeta-function, is used to evaluate some integrals involving the logarithmic derivative of the zeta-function. Some connections are shown between the GUE Hypothesis…
A fresh approach to the long debated question is proposed, starting from the GRAM-BACKLUND analytical continuation of the Zeta function (G-B Zeta expression). Consideration is given to the symmetric (even-exponent) and anti-symmetric (odd…
A new integral representation for the Riemann zeta function is derived. This representation covers the important region of the complex plane where the real part of the argument of the function lies between 0 and 1. Using this…
In 2016, the first-named author introduced a formulation of the Alternative Hypothesis that assumes that consecutive zeros of the Riemann zeta-function are spaced at multiples of half of the average spacing, but does not assume that the…