We have given some arguments that a two-dimensional Lorentz-invariant Hamiltonian may be relevant to the Riemann hypothesis concerning zero points of the Riemann zeta function. Some eigenfunction of the Hamiltonian corresponding to infinite-dimensional representation of the Lorentz group have many interesting properties. Especially, a relationship exists between the zero zeta function condition...

In this note we prove a relation between the Riemann Zeta function, ζ and the ξ function (Krein spectral shift) associated with the Harmonic Oscillator in one dimension. This gives a new integral representation of the zeta function and also a reformulation of the Riemann hypothesis as a question in L1(R). ∗Part of talk presented at the Conference on Harmonic Analysis, 13-15 March 1997, Ramanuja...

and extended analytically to the whole complex plane by a functional equation (see [8, p. 14]). The original Riemann hypothesis asserts that the nonreal zeros of the Riemann zeta function ζ(s) all lie on the line Re(s) = 1/2. In his monumental paper [11] of 1859, Riemann made this assertion in order to derive an expression for the deviation of the exact number of primes ≤ x, which is denoted by...

1. In t roduc t ion . Several recently invented number-theoretic algorithms are sketched below. They all have the common feature that they rely on bounded precision computations of analytic functions. The main one of these algorithms is a new method of calculating values of the Riemann zeta function at multiple points. This method enables one to verify the truth of the Riemann Hypothesis (RH) f...

Hafner and Stopple proved a conjecture of Zagier that the inverse Mellin transform symmetric square [Formula: see text]-function associated to Ramanujan tau function has an asymptotic expansion in terms nontrivial zeros Riemann zeta text]. Later, Chakraborty et al. extended this phenomenon for any Hecke eigenform over full modular group. In paper, we study Lambert series text] where is text]th ...

In this research-cum-expository article, we aim at presenting a systematic account of some recent developments involving the Riemann Zeta function ζ(s), the Hurwitz (or generalized) Zeta function ζ(s, a), and the Hurwitz-Lerch Zeta function Φ(z, s, a) as well as its various interesting extensions and generalizations. In particular, we begin by looking into the problems associated with the evalu...

Abstract. We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the approximation by products is good in this region, the zeta-function has at most finitely many zeros in it. We then construct a parameterized family of non-a...

Abstract We show that some classical identities about multiplicative functions, and the Riemann zeta-function, may conveniently be interpreted in terms of the addition of ±1 to some alphabets. Résumé Nous montrons que des identités classiques concernant des fonctions multiplicatives de la théorie des nombres, et la fonction zeta de Riemann, s’interprètent commodément en terme de l’opération : “...

In the present paper, we construct an algorithm for the evaluation of real Riemann zeta function ζ(s) for all real s, s > 1, in polynomial time and linear space on Turing machines in Ko–Friedman model. The algorithms is based on a series expansion of real Riemann zeta function ζ(s) (the series globally convergents) and uses algorithms for the evaluation of real function (1 + x) h and hypergeo-m...