Abstract The Riemann zeta distribution, defined as the one whose characteristic function is the normalised Riemann zeta function, is an interesting example of an infinitely divisible distribution. The infinite divisibility of the distribution has been proved with recourse to the Euler product of the Riemann zeta function. In this paper, we look at multiple zeta-star function, which is a multi-d...

1. The Fundamental Theorem of Calculus 1 2. Building the Definite Integral 5 2.1. Upper and Lower Sums 5 2.2. Darboux Integrability 7 2.3. Verification of the Axioms 10 2.4. An Inductive Proof of the Integrability of Continuous Functions 12 3. Further Results on Integration 13 3.1. The oscillation 13 3.2. Discontinuities of Darboux Integrable Functions 14 3.3. A supplement to the Fundamental Th...

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...

Assuming the Riemann Hypothesis, Montgomery and Taylor showed that at least 67.25% of the zeros of the Riemann zeta-function are simple. Using Montgomery and Taylor's argument together with an elementary combinatorial argument, we prove that assuming the Riemann Hypothesis at least 67.275% of the zeros are simple.

From eigensolutions of the harmonic oscillator or Kepler-Coulomb Hamiltonian we extend the functional equation for the Riemann zeta function and develop integral representations for the Riemann xi function that is the completed classical zeta function. A key result provides a basis for generalizing the important Riemann-Siegel integral formula.

In this paper we study the Fuchsian Riemann-Hilbert (inverse monodromy) problem corresponding to Frobenius structures on Hurwitz spaces. We find a solution to this Riemann-Hilbert problem in terms of integrals of certain meromorphic differentials over a basis of an appropriate relative homology space over a Riemann surface, study the corresponding monodromy group and compute the monodromy matri...

In this article we compute a discrete mean value of the derivative of the Riemann zeta function. This mean value will be important for several applications concerning the size of ζ(ρ) where ζ(s) is the Riemann zeta function and ρ is a non-trivial zero of the Riemann zeta function.

This paper describes basic properties of the Riemann zeta function and its generalizations, indicates some of geometric analogies, and presents various formulations of the Riemann hypothesis. It briefly discusses the approach of A. Connes to a “spectral” interpretation of the Riemann zeros via noncommuative geometry, which is treated in detail by Paula Tretkoff [33].

The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-)periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation are given. First, a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision. This formula ...

The idea of this work is to compare a new positive and entropy stable approximate Riemann solver by Francois Bouchut with a state-of the-art algorithm for astrophysical fluid dynamics. We implemented the new Riemann solver into an astrophysical PPM-code, the Prometheus code, and also made a version with a different, more theoretically grounded higher order algorithm than PPM. We present shock t...