Abstract. This paper is concerned with the numerical approximation of the solutions of a macroscopic model for the description of the flow of pedestrians. Solutions of the associated Riemann problem are known to be possibly nonclassical in the sense that the underlying discontinuities may well violate the Lax inequalities, which makes their numerical approximation very sensitive. This study pro...

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.

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].

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...

This subject started with H. L. Montgomery’s famous paper [12], where he introduced a method to study the vertical distribution of the zeros of the Riemann zeta-function. Assuming the Riemann Hypothesis, he calculated the pair correlation of the zeros and from it deduced an estimate for the size of small gaps between the zeros and for the percentage of simple zeros. This paper opened a whole ne...

The Riemann zeta function ζ(s) is defined by ζ(s) = ∑∞ n=1 1 ns for R(s) > 1 and may be extended to a regular function on the whole complex plane excluding 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 has a broad application in every branch of mathematics. The density hypo...

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...

The Riemann Zeta Function, ζ(s), is an important complex function whose behavior has implications for the distribution of the prime numbers among the natural numbers. Most notably, the still unsolved Riemann Hypothesis, which states that all non-trivial zeros of the zeta function have real part one-half, would imply the most regular distribution of primes possible in the context of current theo...

is called Fuchsian if the N ×N coefficient matrix A(λ) is a rational function of λ whose singularities are simple poles. Each Fuchsian system generates, via analytic continuation of its fundamental solution Ψ (λ) along closed curves, a representation of the fundamental group of the punctured Riemann sphere (punctured at the poles of A(λ)) in the group of N ×N invertible matrices. This represent...