Riemann hypothesis

About: Riemann hypothesis is a research topic. Over the lifetime, 7850 publications have been published within this topic receiving 119605 citations. The topic is also known as: RH & Riemann Hypothesis.

The Theory of the Riemann Zeta-Function

Abstract: The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects of the theory, starting from first principles and probing the function's own challenging theory, with the famous and still unsolved "Riemann hypothesis" at its heart. The second edition has been revised to include descriptions of work done in the last forty years and is updated with many additional references; it will provide stimulating reading for postgraduates and workers in analytic number theory and classical analysis.

Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction

Abstract: The Equations of Fluid Dynamics.- Notions on Hyperbolic Partial Differential Equations.- Some Properties of the Euler Equations.- The Riemann Problem for the Euler Equations.- Notions on Numerical Methods.- The Method of Godunov for Non#x2014 linear Systems.- Random Choice and Related Methods.- Flux Vector Splitting Methods.- Approximate#x2014 State Riemann Solvers.- The HLL and HLLC Riemann Solvers.- The Riemann Solver of Roe.- The Riemann Solver of Osher.- High#x2013 Order and TVD Methods for Scalar Equations.- High#x2013 Order and TVD Schemes for Non#x2013 Linear Systems.- Splitting Schemes for PDEs with Source Terms.- Methods for Multi#x2013 Dimensional PDEs.- Multidimensional Test Problems.- FORCE Fluxes in Multiple Space Dimensions.- The Generalized Riemann Problem.- The ADER Approach.- Concluding Remarks.

A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation

Abstract: In this article we present a new and general approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, when evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves here exclusively to the modified Korteweg-de Vries (MKdV) equation

Zeta Regularization Techniques with Applications

Abstract: Part 1 The Riemann Zeta function: Riemann, Hurwitz, Epstein, Selberg and related zeta functions analytic continuation - practical uses for series summation asymptotic expansion of "zeta". Part 2 Zeta-function regularization of sums over known spectrum: the zeta-function regularization theorem multiple zeta-functions with arbitrary exponents. Part 3 Zeta-function regularization when the spectrum is not known: zeta-function vs heat-kernel regularization small-"t" asymptotic expansion of the heat-kernel. Part 4 The Casimir effect in flat space-time with compact spatial part: simply connected compact manifold with constant curvature the Selberg trace formula for compact hyperbolic manifolds. Part 5 Finite temperature effects for theories defined on compact hyperbolic manifolds: basic formalism for the finite-temperature effective potential the finite-temperature thermodynamic potential for manifolds with a compact spatial part. Part 6 Properties of the chemical potential in higher-dimensional manifolds: the flat-manifold case the constant non-zero curvature case. Part 7 Strings at non-zero temperature and 2d gravity: free energy for the Bosonic string vacuum energy for Torus compactified strings. Part 8 Membranes at non-zero temperatures: supermembrane free energy free energy for the compactified supermembranes and modular invariance and others.