Theta Functions in Complex Analysis and Number Theory

Mixed sums of squares and triangular numbers (III)

Publisher Summary This chapter discusses the problems and results on combinatorial number theory. It discusses number theoretic problems that are of combinatorial nature. Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers along with the properties of objects made out of integers. Integers can be considered either in themselves or as solutions to equations. Questions in number theory are often understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. Real numbers can also be studied in relation to rational numbers.

Starting with a simple formulation accessible to all mathematicians, this second edition is designed to provide a thorough introduction to nonstandard analysis. Nonstandard analysis is now a well-developed, powerful instrument for solving open problems in almost all disciplines of mathematics; it is often used as a secret weapon by those who know the technique. This book illuminates the subject with some of the most striking applications in analysis, topology, functional analysis, probability and stochastic analysis, as well as applications in economics and combinatorial number theory. The first chapter is designed to facilitate the beginner in learning this technique by starting with calculus and basic real analysis. The second chapter provides the reader with the most important tools of nonstandard analysis: the transfer principle, Keisler s internal definition principle, the spill-over principle, and saturation. The remaining chapters of the book study different fields for applications; each begins with a gentle introduction before then exploring solutions to open problems. All chapters within this second edition have been reworked and updated, with several completely new chapters on compactifications and number theory. Nonstandard Analysis for the Working Mathematician will be accessible to both experts and non-experts, and will ultimately provide many new and helpful insights into the enterprise of mathematics

The effective application of probabilistic reasoning in the study of problems in diverse areas is one of the most exciting recent developments in Mathematics. Probabilistic methods turned out to be very powerful in Discrete Mathematics, Analysis, Number Theory and Theoretical Computer Science. The meeting was dedicated to recent developments in these areas, focusing on the investigation of combinatorial problems for random sets and probabilistic methods, on the study of of questions in percolation, on the design and analysis of randomized algorithms and derandomization techniques, and on applications of probabilistic ideas in the study of questions in Combinatorial Number Theory and in Combinatorial Geometry.

Simultaneous systems of representatives and combinatorial number theory

New relationships between questions in combinatorial number theory are shown. The Erdos-Turan conjecture is shown to be equivalent to the finite problem; the Erdos-Turan conjecture for N is shown to follow from that for Z; and the additive number theory of the integers and that of the finite bit strings under bitwise mod 2 addition are contrasted. The dual space technique from coding theory is applied to the integer question, leading to a lower bound on the number of collisions. A subsidiary question in coding theory is raised, which is a refinement of the perfect code theorem. Finally some computations for cyclic codes are given.

From their inception, Siegel modular forms have been studied extensively because of their significance in both automorphic functions in several complex variables and number theory. The comprehensive theory of automorphic forms to subgroups of algebraic groups and the arithmetical theory of modular forms illustrate these two aspects in an illuminating manner. The author's aim is to present a straightforward and easily accessible survey of the main ideas of the theory at an elementary level, providing a sound basis from which the reader can study advanced works and undertake original research. This book is based on lectures given by the author for a number of years and is intended for a one-semester graduate course, though it can also be used profitably for self-study. The only prerequisites are a basic knowledge of algebra, number theory and complex analysis.

In 1973, Shimura (Ann. Math. (2) 97:440–481, 1973) introduced a family of correspondences between modular forms of half-integral weight and modular forms of even integral weight. Earlier, in unpublished work, Selberg explicitly computed a simple case of this correspondence pertaining to those half-integral weight forms which are products of Jacobi’s theta function and level one Hecke eigenforms. Cipra (J. Number Theory 32(1):58–64, 1989) generalized Selberg’s work to cover the Shimura lifts where the Jacobi theta function may be replaced by theta functions attached to Dirichlet characters of prime power modulus, and where the level one Hecke eigenforms are replaced by more generic newforms. Here we generalize Cipra’s results further to cover theta functions of arbitrary Dirichlet characters multiplied by Hecke eigenforms.

A major theme in proof theory consists in classifying the proof strength of mathematical frameworks for reasoning about mathematics. The resulting phase transitions from provability to unprovability are interesting from the foundational as well as from the mathematical point of view. It is very surprising that during these investigations methods from analytic number theory, combinatorial probability theory, complex analysis and Ramsey theory enter the scene. In this abstract we present the underlying research program. We try to explain our intuition about the nature of phase transition results under discussion. It is our objective to show that there are intriguing interrelations between “usual” mathematics and proof theory. The abstract is intended to be accessible to a general mathematical audience. We therefore concentrate on basic examples and ideas without

Given a compact Riemann surface $X$, we consider the line, in the space of holomorphic sections of $2\Theta $ on $J^{0}(X)$, orthogonal to all the sections that vanish at the origin. This line produces a natural meromorphic bidifferential on $X\times X$ with a pole of order two on the diagonal. This bidifferential is extensively investigated. In particular we show that it produces a projective structure on $X$, which is different from the standard ones.

Theta functions have historically played a prominent role in number theory. One such role is the construction of modular forms. In this work, a generalized theta function is used to construct an infinite family of summation identities. Our results grew out of some observations noted during a presentation given by the author at the 1992 AMS-MAA-SIAM Joint Meetings in Baltimore.

Consider a codebook containing N unit-norm complex vectors in a K-dimensional space. In a number of applications, the codebook that minimizes the maximal cross-correlation amplitude (I/sub max/) is often desirable. Relying on tools from combinatorial number theory, we construct analytically optimal codebooks meeting, in certain cases, the Welch lower bound. When analytical constructions are not available, we develop an efficient numerical search method based on a generalized Lloyd algorithm, which leads to considerable improvement on the achieved I/sub max/ over existing alternatives. We also derive a composite lower bound on the minimum achievable I/sub max/ that is effective for any codebook size N.

A retrospective on 60 years of approximation theory and associated fields

We use ergodic-theoretical tools to study various notions of “large” sets of integers which naturally arise in theory of almost periodic functions, combinatorial number theory, and dynamics. Call a subset of N a Bohr set if it corresponds to an open subset in the Bohr compactification, and a piecewise Bohr set (PWB) if it contains arbitrarily large intervals of a fixed Bohr set. For example, we link the notion of PWB-sets to sets of the form A+B, where A and B are sets of integers having positive upper Banach density and obtain the following sharpening of a recent result of Renling Jin.

We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense, close to abelian. We prove two main theorems. In the first, we completely classify all finite groups with an automorphism cubing more than half their elements. All such groups are either nilpotent class 2 or possess an abelian subgroup of index 2. For our second theorem, we show that if a group possesses an automorphism sending more than 4/15 of its elements to their cubes, then it must be solvable. The group A_5 shows that this result is best-possible. Both our main findings closely parallel results of previous authors on finite groups possessing an automorphism which inverts many elements. The technicalities of the new proofs are somewhat more subtle, and also throw up a nice connection to a basic problem in combinatorial number theory, namely the study of subsets of finite cyclic groups which avoid non-trivial solutions to one or more translation invariant linear equations.