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...

The computation of the complex zeros of an incomplete Riemann zeta function defined in an earlier paper is extended and new zero trajectories are given. A second incomplete Riemann zeta function is denned and its zero trajectories are investigated numerically as functions of the upper limit X of the definition integral. It becomes apparent that there exist three different classes of zero trajec...

In 1972, H.L. Montgomery and F. Dyson uncovered a surprising connection between the Theory of the Riemann Zeta function and Random Matrix Theory. For the next few decades, the major developments in the area were the numerical calculations of Odlyzko and conjectures for the moments of the Riemann Zeta function (and other L-functions) found by Conrey, Ghosh, Gonek, Heath-Brown, Hejhal and Sarnak....

The Riemann zeta function ζ(s) is defined by ζ(s) = ∑∞ n=1 1 ns for R(s) > 1 and can be extended to a regular function on the whole complex plane deleting its unique pole at s = 1. The Riemann hypothesis is a conjecture made by Riemann in 1859 asserting that all non-trivial zeros for ζ(s) lie on the line R(s) = 12 , which is equivalent to the prime number theorem in the form of π(x)−Li(x) = O(x...

0 |ζ( 1 2 + it)| dt. For positive real numbers k, it is believed that Mk(T ) ∼ CkT (logT ) 2 for a positive constant Ck. A precise value for Ck was conjectured by Keating and Snaith [9] based on considerations from random matrix theory. Subsequently, an alternative approach, based on multiple Dirichlet series and producing the same conjecture, was given by Diaconu, Goldfeld and Hoffstein [4]. R...

c © W W L Chen, 1990, 2013. This chapter originates from material used by the author at Imperial College London between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage an...

The multiindex Mittag-Leffler (M-L) function and the multiindex Dzrbashjan-Gelfond-Leontiev (D-G-L) differentiation and integration play a very pivotal role in the theory and applications of generalized fractional calculus. The object of this paper is to investigate the relations that exist between the Riemann-Liouville fractional calculus and multiindex Dzrbashjan-Gelfond-Leontiev differentiat...

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...

The Riemann Hypothesis, which specifies the location of zeros of the Riemann zeta function, and thus describes the behavior of primes, is one of the most famous unsolved problems in mathematics, and extensive efforts have been made over more than a century to check it numerically for large sets of cases. Recently a new algorithm, invented by the speaker and A. Scho . . nhage, has been implement...