Abstract: # Finsler Metrics # Structure Equations # Geodesics # Parallel Translations # S-Curvature # Riemann Curvature # Finsler Metrics of Scalar Flag Curvature # Projectively Flat Finsler Metrics

Abstract: In August 1859 Bernhard Riemann, a little-known 32-year old mathematician, presented a paper to the Berlin Academy titled: "On the Number of Prime Numbers Less Than a Given Quantity." In the middle of that paper, Riemann made an incidental remark a " a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years. Today, after 150 years of careful research and exhaustive study, the question remains. Is the hypothesis true or false? Riemann's basic inquiry, the primary topic of his paper, concerned a straightforward but nevertheless important matter of arithmetic a " defining a precise formula to track and identify the occurrence of prime numbers. But it is that incidental remark a " the Riemann Hypothesis a " that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant shrouded in the shadows a " subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age. It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp, holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. Hunting down the solution to the Riemann Hypothesis has become an obsession for many a " the veritable "great white whale" of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution. Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed biography and history, Prime Obsession is a fascinating and fluent account of an epic mathematical mystery that continues to challenge and excite the world. Posited a century and a half ago, the Riemann Hypothesis is an intellectual feast for the cognoscenti and the curious alike. Not just a story of numbers and calculations, Prime Obsession is the engrossing tale of a relentless hunt for an elusive proof a " and those who have been consumed by it.

Abstract: The random matrix model predicts that many statistics associated to zeros of a family of L-functions can be modeled (or predicted) by the distribution of eigenval ues of large random matrices in one of the classical linear groups If the statistics of a family of L-functions are modeled by the eigenvalues of the group G, then we say that G is the symmetry group (or symmetry type) associated to the family The statistic of interest to us in this work is the density of zeros near the central point (also known as the 1-level density) The random matrix model predicts that the distribution of these zeros should be modeled by the eigenvalues nearest 1 for one of the symmetry types G (unitary, symplectic, and orthogonal) All of the different groups G have distinct behavior in this regard Therefore, computing the 1-level density gives a theoretical way to predict the symmetry type of a family The 1-level density has been studied for a wide variety of families of L-functions; see [R], [KSI], [ILS], [Mil] for example It is standard to assume the Generalized Riemann Hypothesis (GRH) to study the 1-level density and we do so throughout this work In particular, it is necessary to use GRH for the application of obtaining a bound on the average analytic rank from a density theorem In some cases the use of GRH improves the range of the density theorem, which translates to an improved bound on the average rank Besides these crucial applications of GRH, we have freely assumed GRH even when it could be removed with extra work since it simplifies arguments in some non essential places It is especially interesting to investigate the 1-level density for families of L functions attached to elliptic curves over the rationals since zeros at the central point have important arithmetic information (by the conjecture of Birch and Swinnerton Dyer) These investigations have been the main focus of this work

Abstract: We generalize the logarithmic version of the Riemann- Hilbert correspondence defined in (KtNk) to local systems with quasi- unipotent local monodromies by working with a certain Grothendieck topology. We also discuss its behavior with respect to direct images and give applications to nearby cycles and the degeneration of relative log Hodge to log de Rham spectral sequences.

Abstract: Given a vector bundle $F$ on a smooth Deligne-Mumford stack $\X$ and an invertible multiplicative characteristic class $\bc$, we define the orbifold Gromov-Witten invariants of $\X$ twisted by $F$ and $\bc$. We prove a "quantum Riemann-Roch theorem" which expresses the generating function of the twisted invariants in terms of the generating function of the untwisted invariants. A Quantum Lefschetz Hyperplane Theorem is derived from this by specializing to genus zero. As an application, we determine the relationship between genus-0 orbifold Gromov-Witten invariants of $\X$ and that of a complete intersection. This provides a way to verify mirror symmetry predictions for complete intersection orbifolds.

Abstract: We study the distribution of zeros of holomorphic modular forms. Assuming the Generalized Riemann Hypothesis we show that the zeros of Hecke eigenforms for the modular group become equidistributed with respect to the hyperbolic measure on the modular domain as the weight grows.

Abstract: We study the distribution of zeros of holomorphic modular forms. Assuming the Generalized Riemann Hypothesis we show that the zeros of Hecke eigenforms for the modular group become equidistributed with respect to the hyperbolic measure on the modular domain as the weight grows.

Abstract: A Brief History of Prime.- Diophantine Equations.- Quadratic Diophantine Equations.- Recovering the Fundamental Theorem of Arithmetic.- Elliptic Curves.- Elliptic Functions.- Heights.- The Riemann Zeta Function.- The Functional Equation of the Riemann Zeta Function.- Primes in an Arithmetic Progression.- Converging Streams.- Computational Number Theory.

Abstract: In this paper we give some background theory on the concept of fractional calculus, in particular the Riemann-Liouville operators. We then investigate the Taylor-Riemann series using Osler’s theorem and obtain certain double infinite series expansions of some elementary functions. In the process of this we give a proof of the convergence of an alternative form of Heaviside’s series. A Semi-Taylor series is introduced as the special case of the Taylor-Riemann series when α = 1/2, and some of its relations to special functions are obtained via certain generating functions arising in complex fractional calculus.

Abstract: The aim of the present work is to exhibit a new proof of the explicit spectral expansion for the fourth moment of the Riemann zeta-function that was established by the second named author a decade ago. Our proof is new, particularly in the sense that it dispenses completely with the Kloostermania, the spectral theory of sums of Kloosterman sums that was used in the former proof. The argument is now constructed precisely upon the spectral structure of the Lie group PSL(2,R). Main ingredients in our argument are the theory of automorphic representations as well as the harmonic analysis on the big Bruhat cell. In essence, this work of ours indicates a new way to view the Riemann zeta-function.

Abstract: This volume focuses on various aspects of zeta functions: multiple zeta values, Ohno’s relations, the Riemann hypothesis, L-functions, polylogarithms, and their interplay with other disciplines. Eleven articles on recent advances are written by outstanding experts in the above-mentioned fields. Each article starts with an introductory survey leading to the exciting new research developments accomplished by the contributors. This book will become the major standard reference on the recent advances on zeta functions.

Abstract: We give explicit formulae for the local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically satisfied by groups with small centre and sufficiently large abelianization. We show how the functional equations of two types of zeta functions – the Weil zeta function associated to an algebraic variety and zeta functions of algebraic groups introduced by Igusa – match up to give a functional equation for local normal zeta functions of groups. We also give explicit formulae and derive functional equations for an infinite family of class-2-nilpotent groups known as Grenham groups, confirming conjectures of du Sautoy.

Abstract: We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence {λk}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence ηj are valid. The constants ηj enter the Laurent expansion of the logarithmic derivative of the zeta function about s=1 and appear to have remarkable characteristics. On our conjecture, not only does the Riemann hypothesis follow, but an inequality governing the values λn and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function.

Abstract: We use a smoothed version of the explicit formula to find an approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function that involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory.

Abstract: We prove the Hodge-Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kahler manifolds in the mixed situation.

Abstract: Mathematics Subject Classification (2000): Primary 11M26; Secondary 11K38 We continue our investigation of the distribution of the fractional parts of �, whereis a fixed non-zero real number and runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to Mont- gomery's pair correlation function and the distribution of primes in short intervals. We also discuss analogous results for a more general L-function.

Abstract: We give an explicit representation for the sums of multiple zeta-star values of fixed weight and height in terms of Riemann zeta values.

Abstract: We propose some numerical tests for identifying L-functions of automorphic representations of GL(r) over a number field. We then apply the tests to various conjectured automorphic L-functions, provid- ing evidence for their modularity and the associated Riemann hypotheses. Our chief examples are the Hasse-Weil L-functions attached to curves of genus 2 over Q and to elliptic curves over Q( √ −1). We discuss also three miscellaneous applications. The first two include the L-functions of high symmetric powers of Ramanujan'sand the modular form in S2(� 0(11)). The third application is an even 2-dimensional icosahedral Galois repre- sentation over Q, which conjecturally corresponds to a Maass form of eigenvalue 1 .

Abstract: The paper studies the local zero spacings of deformations of the Riemann ξ-function under certain averaging and differencing operations. For real h we consider the entire functions Ah(s) := 1 (ξ(s + h) + ξ(s − h)) and Bh(s) = 1 2i (ξ(s + h) − ξ(s − h)) . For |h| ≥ 1 2 the zeros of Ah(s) and Bh(s) all lie on the critical line ℜ(s) = 1 and are simple zeros. The number of zeros of these functions to height T has asymptotically the same density as the Riemann zeta zeros. For fixed |h| ≥ 1 the distribution of normalized zero spacings of these functions up to height T converge as T → ∞ to a limiting distribution, which consists of equal spacings of size 1. That is, these zeros are asymptotically regularly spaced. Assuming the Riemann hypothesis, the same properties hold for all nonzero h. In particular, these averaging and differencing operations destroy the (conjectured) GUE distribution of the zeros of the ξ-function, which should hold at h = 0. Analogous results hold for all completed Dirichlet L-functions ξχ(s) having χ a primitive character.

Abstract: The double zeta function was first studied by Euler in response to a letter from Goldbach in 1742. One of Euler's results for this function is a decomposition formula, which expresses the product of two values of the Riemann zeta function as a finite sum of double zeta values involving binomial coefficients. Here, we establish a q-analog of Euler's decomposition formula. More specifically, we show that Euler's decomposition formula can be extended to what might be referred to as a “double q-zeta function” in such a way that Euler's formula is recovered in the limit as q tends to 1.

Abstract: Summations and relations involving the Hurwitz and Riemann ζ-functions are extended first to Barnes ζ-functions and then to ζ-functions of general type. The analysis is motivated by the evaluation of determinants on spheres which are treated both by a direct expansion method and by regularized sums. Comments on existing calculations are made. It is suggested that the combination ζ′(-n) + Hnζ(-n), where Hn is a harmonic number, should be taken as more relevant than just ζ′(-n). This leads to a Kaluza–Klein technique, providing a determinant interpretation of the Glaisher–Kinkelin–Bendersky constants which are then generalized to arbitrary ζ-functions. This technique allows an improved treatment of sphere determinants.

Abstract: In this article, Riemann boundary value problems (BVPs) of polyanalytic functions and metaanalytic functions on the closed curve are investigated. Using the theorem of decomposition of polyanalytic functions, the expression of solution and the condition of solvability for Riemann BVP of polyanalytic functions are obtained by reducing the problem to the equivalent Riemann BVP of analytic functions. Then the expression of solution and the condition of solvability for Riemann BVP of metaanalytic functions are obtained by reducing the problem into the equivalent Riemann BVP of polyanalytic functions.

Abstract: We construct higher genus Riemann's minimal surfaces properly embedded in the Euclidean space. To do that we glue end by end a Costa-Hoffman-Meeks examples to two halves genus zero Riemann's minimal surfaces. In first we need to perform a deformation of a Costa-Hoffman-Meeks example to prescribe the flux vector along the catenoidal ends. Then we study the mapping property of the Jacobi operator on the half Riemann example as a perturbation analysis of a CMC-Delaunay half cylinder.