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...

Abstract. By the use of techniques of real analysis and weight functions, we obtain two lemmas and build a few equivalent conditions of a Hardy-type integral inequality with a non-homogeneous kernel, related to a parameter where the constant factor is expressed in terms of the extended Riemann zeta function. Meanwhile, a few equivalent conditions for two kinds of Hardytype integral inequalities...

The explicit formulas of Riemann and Guinand-Weil relate the set of prime numbers with the set of nontrivial zeros of the zeta function of Riemann. We recall Alain Connes’ spectral interpretation of the critical zeros of the Riemann zeta function as eigenvalues of the absorption spectrum of an unbounded operator in a suitable Hilbert space. We then give a spectral interpretation of the zeros of...

In this article we derive a generalization of the Riemann-Siegel asymptotic formula for the Riemann zeta function. By subtracting the singularities closest to the critical point we obtain a significant reduction of the error term at the expense of a few evaluations of the error function. We illustrate the efficiency of this method by comparing it to the classical Riemann-Siegel formula.

The Stieltjes constants γk(a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about its only pole at s = 1. We present the relation of γk(1) to the ηj coefficients that appear in the Laurent expansion of the logarithmic derivative of the Riemann zeta function about its pole at s = 1. We obtain novel integral representations of the Stieltjes constants and new d...

Riemann hypothesis states that the nontrivial zeros of Riemann Zeta function lie on the axis x = 1/2. Since Riemann zeta function allows interpretation as a thermodynamical partition function for a quantum field theoretical system consisting of bosons labelled by primes, it is interesting to look Riemann hypothesis from the perspective of physics. Quantum TGD and also TGD inspired theory of con...

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 of items, in this paper, we look at the Riemann Hypothesis using a new applied approach to infinity allowing one to easily execute numerical computations with...

In 1999, Iwan Duursma defined the zeta function for a linear code as a generating function of its Hamming weight enumerator. It can also be defined for other homogeneous polynomials not corresponding to existing codes. If the homogeneous polynomial is invariant under the MacWilliams transform, then its zeta function satisfies a functional equation and we can formulate an analogue of the Riemann...

In this paper, we derive a more precise version of the Strong Pair Correlation Conjecture on the zeros of the Riemann zeta function under Riemann Hypothesis and Twin Prime Conjecture.