Abstract: The prime number theorem, that the number of primes < x is asymptotic to x/log x, was proved (independently) by Hadamard and de la Vallee Poussin in 1896. Their proof had two elements: showing that Riemann's zeta function ;(s) has no zeros with Sc(s) = 1, and deducing the prime number theorem from this. An ingenious short proof of the first assertion was found soon afterwards by the same authors and by Mertens and is reproduced here, but the deduction of the prime number theorem continued to involve difficult analysis. A proof that was elementaty in a technical sense-it avoided the use of complex analysis-was found in 1949 by Selberg and Erdos, but this proof is very intricate and much less clearly motivated than the analytic one. A few years ago, however, D. J. Newman found a very simple version of the Tauberian argument needed for an analytic proof of the prime number theorem. We describe the resulting proof, which has a beautifully simple structure and uses hardly anything beyond Cauchy's theorem. Recall that the notation f(x) g(x) ("f and g are asymptotically equal")

Abstract: The Selberg trace formula on a Riemann surface X connects the discrete spectrum of the Laplacian with the length spectrum of the surface, that is, the set of lengths of the closed geodesics of on X. The connection is most strikingly expressed in terms of the Selberg zeta function, which is a meromorphic function of a complex variable s that is defined for <(s) > 1 in terms of the length spectrum and that has zeros at all s ∈ C for which s(1 − s) is an eigenvalue of the Laplacian in L(X). We will be interested in the case when X is the quotient of the upper half-plane H by either the modular group Γ1 = SL(2,Z) or the extended modular group Γ = GL(2,Z), where γ = ( a b c d ) ∈ Γ acts on H by z 7→ (az + b)/(cz + d) if det(γ) = +1 and z 7→ (az̄ + b)/(cz̄ + d) if det(γ) = −1. In this case the length spectrum of X is given in terms of class numbers and units of orders in real quadratic fields, while the eigenfunctions of the Laplace operator are the non-holomorphic modular functions usually called Maass wave forms. (Good expositions of this subject can be found in [6] and [7]).

Abstract: The Kadomtsev–Petviashvili (KP) equation is known to admit explicit periodic and quasiperiodic solutions with N independent phases, for any integer N, based on a Riemann theta-function of N variables. For N=1 and 2, these solutions have been used successfully in physical applications. This article addresses mathematical problems that arise in the computation of theta-functions of three variables and with the corresponding solutions of the KP equation. We identify a set of parameters and their corresponding ranges, such that every real-valued, smooth KP solution associated with a Riemann theta-function of three variables corresponds to exactly one choice of these parameters in the proper range. Our results are embodied in a program that computes these solutions efficiently and that is available to the reader. We also discuss some properties of three-phase solutions.

Abstract: 1 The exceptional set for Goldbach's problem in short intervals R C Baker, G Harman and J Pintz 2 On an additive property of stable sets A Balog and I Rusza 3 Squarefree values of polynomials and the abc-conjecture J Browkin, M Filaseta, G Greaves and A Schinzel 4 The values of binary linear forms at prime arguments J Brudern, R J Cook and A Perelli 5 Some applications of sieves of dimension exceeding 1 H Diamond and H Halberstam 6 Representations by the determinant and mean values of L-functions W Duke, J Friedlander and H Iwaniec 7 On Montgomery-Hooley asymptotic formula D A Goldston and R C Vaughan 8 Franel integrals R R Hall 9 Eratosthenes, Legendre, Vinogradov and beyond G Harman 10 On hypothesis K* in Waring's problem C Hooley 11 Moments of differences between square-free numbers M N Huxley 12 On the ternary additive problem and the sixth moment of the zeta-function A Ivic 13 A variant of the circle method M Jutila 14 The resemblance of the behaviour of the remainder terms Es(t), D1-2s(x) and R(s+it) I Kiuchi and K Matsumoto 15 A note on the number of divisors of quadratic polynomials J McKee 16 On the distribution of integer points in the real locus of an affine toric variety B Z Moroz 17 An asymptotic expansion of the square of the Riemann zeta-function Y Motohashi 18 The mean square of Dedekind zeta-functions of quadratic number fields Y Motohashi 19 Artin's conjecture and elliptic analogues M R Murty

Abstract: We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation

Abstract: We give a proof of a conjecture of P. Schapira and J.-P. Schneiders on the characteristic classes of D-modules.

Abstract: For the purpose of computing a Godunov flux, Harten, Lax and van Leer [148] presented a novel approach for solving the Riemann problem approximately. The resulting Riemann solvers have become known as HLL Riemann solvers. In this approach an approximation for the intercell numerical flux is obtained directly, unlike the Riemann solvers presented previously in Chaps. 4 and 9. The central idea is to assume a wave configuration for the solution that consists of two waves separating three constant states. Assuming that the wave speeds are given by some algorithm, application of the integral form of the conservation laws gives a closed-form, approximate expression for the flux. The approach produced practical schemes after the contributions of Davis [94] and Einfeldt [105], who independently proposed various ways of computing the wave speeds required to completely determine the intercell flux. The resulting HLL Riemann solvers form the bases of very efficient and robust approximate Godunov-type methods. One difficulty with these schemes, however, is the assumption of a two-wave configuration. This is correct only for hyperbolic systems of two equations, such as the one-dimensional shallow water equations. For larger systems, such as the Euler equations or the split two-dimensional shallow water equations for example, the two-wave assumption is incorrect. As a consequence the resolution of physical features such as contact surfaces, shear waves and material interfaces, can be very inaccurate. For the limiting case in which these features are stationary relative to the mesh, the resulting numerical smearing is unacceptable. In view of these shortcomings of the HLL approach, a modification called the HLLC Riemann solver (C stands for Contact) was put forward by Toro, Spruce and Speares [347]. In spite of the limited experience available in using the HLLC scheme, the evidence is that this appears to offer a useful approximate Riemann solver for practical applications. Batten, Goldberg and Leschziner [18] have recently proposed implicit versions of the HLLC Riemann solver, and have applied the schemes to turbulent flows.

Abstract: We establish functional equations for peculiar functionsf:I → ℝ,I ⊂ ℝ an interval, such as (1) continuous, nowhere differentiable functions of various types (Weierstrass, Takagi, Knopp, Wunderlich), (2) Riemann's function, which is nondifferentiable except on certain rational points, (3) singular functions of various types (Cantor, Minkowski, de Rham).

Abstract: Let p be an odd prime. Assuming the Extended Riemann Hypothesis, we show how to construct O((logp) 4 (loglogp) -3 ) residues modulo p, one of which must be a primitive root, in deterministic polynomial time. Granting some well-known character sum bounds, the proof is elementary, leading to an explicit algorithm.

Abstract: We assume the generalized Riemann hypothesis and prove an asymptotic formula for the number of primes for which F p * can be generated by r given multiplicatively independent numbers. In the case when the r given numbers are primes, we express the density as an Euler product and apply this to a conjecture of Brown-Zassenhaus (J. Number Theory 3 (1971), 306-309). Finally, in some examples, we compare the densities approximated with the natural densities calculated with primes up to 9 10 4 .

Abstract: Riemann’s z function perhaps first appeared in statistical mechanics in 1900 in Planck’s theory of the blackbody radiation and then in 1912 in Debye’s theory of the specific heats of solids @1#. Subsequently, this function has played an important role in the statistical theory of the ideal Bose gas, especially for the understanding of Bose-Einstein condensation ~BEC !@ 2 #. More recently, this function together with the Mellin transform has become a powerful tool for the analysis of the thermodynamic potentials @3,4#. It would be no surprise to find fruitful applications of Riemann’s z function in other areas of today’s theoretical physics @5#. Recently it was found that the statistical thermodynamics of ideal gases can be given a unified picture through polylogs defined in terms of the fugacity z and dimensions d @6#. There is richness that this unified picture reveals, such as the anomalous physics in null dimension @7#, the Fermi-Bose reflection in d>3 @8#, and the Fermi-Bose equivalence in d 52 @9#. These physical results are consequences of some special properties of polylogs. It has been long known that a polylog of integral order becomes Riemann’s z function of the same order when its argument attains unity @10#. Thus Riemann’s z function can enter into the unified theory of the statistical thermodynamics via the polylogs quite naturally. Interestingly, we find that this formulation shows another way of evaluating Riemann’s z function, which is presented in this work. The classical theory of polylogs begins with Euler’s dilog and Landen’s trilog, and extends to higher order polylogs such as the quadrilog. In physical applications the order of a polylog is related to physical dimensions d. Thus polylogs of integral order lower than the dilog have been conceived, such as the nil-log and the monolog for the physics in d 50 and 2 @6#. There have been suggestions that negative dimensions can be of theoretical interest @11#. They require the polylogs of still lower orders than the nil-log, departing from the direction of the classical theory of polylogs. What we find is that in these circumstances there exists even a simpler relationship between the polylogs and Riemann’s z function. We can use this relationship to evaluate Riemann’s z function very simply, perhaps more simply than by most standard methods. It also lends an interesting insight into the nature of the values of Riemann’s z function. II. POLYLOGS AND THEIR PROPERTIES

Abstract: This is just a collection of my notes on Riemann normal coordinates.

Abstract: A consistent procedure for regularization of divergences and for the subsequent renormalization of the string tension is proposed in the framework of the one-loop calculation of the interquark potential generated by the Polyakov-Kleinert string. In this way, a justification of the formal treatment of divergences by analytic continuation of the Riemann and Epstein-Hurwitz zeta functions is given. A spectral representation for the renormalized string energy at zero temperature is derived, which enables one to find the Casimir energy in this string model at nonzero temperature very easy.

Abstract: The paper studies the Riemann problem for a conservation law with a source term and a nonconvex f{l}ux-function. The complete solution is provided in the case when the f{l}ux has one inf{l}ection point and the Riemann states are stationary states of the source term. For small times, the structure of the solutions is similar to the homogeneous case. As the time increases, the size of the shocks may decrease under the action of the source, while rarefaction waves tend to traveling waves. It is also proved that if the f{l}ux has more than one inf{l}ection point, there may be shocks vanishing in finite time, in contrast to the case when the f{l}ux is convex.

Abstract: A well-known conjecture of E. Artin [1] states that for any integers $a e {\pm}1$ and $a$ is not a perfect square, there are infinitely many prime integers $p$ for which $a$ is a primitive root $({\bmod}\, p)$ . An analogue of this conjecture for function fields was attacked successfully by Bilharz [2] in 1937 using the Riemann hypothesis for curves over finite fields (subsequently proved by A. Weil). The original conjecture of Artin remains open, though it was shown to be true if one assumes the Generalized Riemann hypothesis by Hooley [7]. In recent years, this conjecture of Artin has also been formulated and studied for elliptic curves over global fields instead of just ${\rm G}_m$ (the original case) (see [11]).

Abstract: The Hamilton-Jacobi equation describes the dynamics of a hypersurface in \(\). This equation is a nonlinear conservation law and thus has discontinuous solutions. The dependent variable is a surface gradient and the discontinuity is a surface cusp. Here we investigate the intersection of cusp hypersurfaces. These intersections define (n-1)-dimensional Riemann problems for the Hamilton-Jacobi equation. We propose the class of Hamilton-Jacobi equations as a natural higher-dimensional generalization of scalar equations which allow a satisfactory theory of higher-dimensional Riemann problems. The fist main result of this paper is a general framwork for the study of higher-dimensional Riemann problems for Hamilton-Jacobi equations. The purpose of the framwork ist to unterstand the structure of Hamilton-Jacobi wave interactions in an explicit and constructive manner. Specialized to two-dimensional Riemann problems (i.e., the intersection of cusp curves on surfaces embedded in \(\)), this framework provides explicit solutions to a number of cases of interest. We are specifically interested in models of deposition and etching, important processes for the manufacture of semiconductor chips.

Abstract: 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 and the absence of trivial representations in the wave function.

Abstract: A new lattice based scheme for numerical relativity will be presented. The scheme uses the same data as would be used in the Regge calculus (eg. a set of leg lengths on a simplicial lattice) but it differs significantly in the way that the field equations are computed. In the new method the standard Einstein field equations are applied directly to the lattice. This is done by using locally defined Riemann normal coordinates to interpolate a smooth metric over local groups of cells of the lattice. Results for the time symmetric initial data for the Schwarzschild spacetime will be presented. It will be shown that the scheme yields second order accurate estimates (in the lattice spacing) for the metric and the curvature. It will also be shown that the Bianchi identities play an essential role in the construction of the Schwarzschild initial data.