Related papers: On a function of Marcel Riesz

New integral formulas involving the Meijer $G$-function are derived using recent results concerning distributional characterisations and distributional transformations in probability theory.

The statement of the Riemann hypothesis makes sense for all global fields, not just the rational numbers. For function fields, it has a natural restatement in terms of the associated curve. Weil's work on the Riemann hypothesis for curves…

The Fejer-Riesz theorem has inspired numerous generalizations in one and several variables, and for matrix- and operator-valued functions. This paper is a survey of some old and recent topics that center around Rosenblum's operator…

This note concerns the general formulation by Preiss and Uher of Kestelman's influential result pertaining the change of variable, or substitution, formula for the Riemann integral.

In this paper, we give a short elementary proof of the well known Euler's recurrence formula for the Riemann zeta function at positive even integers and integral representations of the Riemann zeta function at positive integers and at…

The Riemann Hypothesis has been of central interest to mathematicians for a long time and many unsuccessful attempts have been made to either prove or disprove it. Since the Riemann zeta function is defined as a sum of the infinite number…

We introduce an explicit formula for a reciprocal sum related to the Riemann zeta function at s=6, and pose one question related to a computational formula for larger values of s.

We establish effective mean-value estimates for a wide class of multiplicative arithmetic functions, thereby providing (essentially optimal) quantitative versions of Wirsing's classical estimates and extending those of Hal\'asz. Several…

Analyzing in detail the analytic continuation of the Riemann zeta function we are able to generate several new identities which may be useful for application in physics and mathematics.

We develop the $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic twist of modular $L$-functions using multiple Dirichlet series under the generalized Riemann…

We present a set of lectures on topics of advanced calculus in one real and complex variable with several new results and proofs on the subject, specially with detailed proof-always missing in the literature - of the Cissoti explicitly…

In this paper we set up the theory of acid zeta function and ajoint acid zeta function, based on the theory, we point out a reason to doubt the truth of the Riemann hypothesis and also as a consequence, we give out some new RH equivalences.

The Riemann Hypothesis is reformulated as statements about eigenvalues of some matrices entries of which are defined via Taylor coefficient of the zeta function. These eigenvalues demonstrate interesting visual patterns allowing one to…

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

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 $\nu$,…

In this note a characterization of anallytically Riesz operators is given. This work completes the article [1].

In this paper we have two issues coming from the same background. The first one is to describe a certain ratio of Fredholm determinants of integral operators arising from the Riemann zeta function by using the solution of a single integral…

We use integration by parts formulas to give estimates for the $L^p$ norm of the Riesz transform. This is motivated by the representation formula for conditional expectations of functionals on the Wiener space already given in Malliavin and…

This paper is a continuation of our recent paper with the same title, arXiv:0806.1596v1 [math.NT], where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that…

In the paper, we first prove a sufficient condition for the Riemann hypothesis which involves the order of magnitude of the partial sum of the Liouville function. Then we show a formula which is curiously related to the proved sufficient…