Based on work of Alain Connes, I have constructed a spectral interpretation for zeros of L-functions. Here we specialise this construction to the Riemann ζ-function. We construct an operator on a nuclear Fréchet space whose spectrum is the set of non-trivial zeros of ζ. We exhibit the explicit formula for the zeros of the Riemann ζ-function as a character formula.

The function S(T) is the error term in the formula for the number of zeros of the Riemann zeta-function above the real axis and up to height Tin the complex plane. We assume the Riemann hypothesis, and examine how well S(T) can be approximated by a Dirichlet polynomial in the Lz norm.

In this paper we introduce an abstract integral which includes both product integrals and additive integrals. Thus our theory of Riemann integration embraces both the product integrals of Volterra [l, 2, 3](2), Birkhoff [3], and Masani [l] and the classical additive integrals of Riemann [l] and Stieltjes [l]. Similarly our theory of Lebesgue integration includes the Lebesgue product integrals o...

This thesis consists of three contributions to the theory of complex approximation on Riemann surfaces. It is known that if E is a closed subset of an open Riemann surface R and f is a holomorphic function on a neighbourhood of E, then it is “usually” not possible to approximate f uniformly by functions holomorphic on all of R. In Chapter 2, we show, however, that for every open Riemann surface...

Abstract Dirichlet series associated with the Poincaré attached to $$\mathrm{SL}(2,{{\mathbb {Z}}})$$ SL ( 2 , Z ) are introduced. Integral representations and transformation formulas given, which descr...

This article introduces a new application of piecewise linear orthogonal triangular functions to solve fractional order differential-algebraic equations. The generalized triangular function operational matrices for approximating Riemann-Liouville fractional order integral in the triangular function (TF) domain are derived. Error analysis is carried out to estimate the upper bound of absolute er...

These notes are a rather subjective account of the theory of dynamical zeta functions. They correspond to three lectures presented by the author at the “Numeration” meeting in Leiden in 2010. 1 A Selection of Zeta Functions In its various manifestations, a zeta function ζ(s) is usually a function of a complex variable s ∈ C. We will concentrate on three main types of zeta function, arising in t...

We introduce a screw function corresponding to the Riemann zeta-function and study its properties from various aspects. Typical results are several equivalent conditions for hypothesis in terms of function. One them can be considered an analog so-called Weil's positivity or Li's criterion. In addition, we prove few partial but unconditional such equivalents.

The quantum mechanics of a system of charged particles interacting with a magnetic field on Riemann surfaces is studied. We explicitly construct the wave functions of ground states in the case of a metric proportional to the Chern form of the θ-bundle, and the wave functions of the Landau levels in the case of the the Poincaré metric. The degeneracy of the the Landau levels is obtained by using...

We describe methods for the derivation of strong asymptotics for the denominator polynomials and the remainder of Padé approximants for a Markov function with a complex and varying weight. Two approaches, both based on a Riemann-Hilbert problem, are presented. The first method uses a scalar Riemann-Hilbert boundary value problem on a two-sheeted Riemann surface, the second approach uses a matri...