Number Theory

Among the thousands of discoveries made by mathematicians over the centuries, some stand out as significant landmarks. One of these is the prime number theorem, which describes the asymptotic distribution of prime numbers. It can be stated in various equivalent forms, two of which are: and

Here we shall survey the developments regarding two well known problems in Geometry of Numbers. The first is a conjecture of Minkowski about the product of non-homogeneous real linear forms. The second one is a conjecture of Watson concerning non-homogeneous real indefinite quadratic forms. Whereas the first one is still resisting solution in the general case, the second one has been completely proved.

Our theme is a relation between the sign of a real function and the analytic behaviour of its associated generating function at a special point on the boundary of convergence. The central idea is the plausible principle: Let f denote a real valued function with support contained in [0, ∞) or the nonnegative integers and let f̂ denote the generating function associated with f. If f is ultimately of one sign, then extremal behaviour of | f̂ | occurs around the real point with largest real part on the boundary of the region of convergence of the series or integral that defines f̂.

The aim of this article is to give an exposition of certain applications of the study of the homogeneous space SL(n, ℝ)/SL(n, ℤ) and the flows on it induced by subgroups of SL(n, ℝ), to problems on values of linear and quadratic forms at integral points. Also, some complements to Margulis’s theorem on Oppenheim’s conjecture are proved.

In Metric Number Theory we are concerned with the arithmetical properties of almost all numbers (in the first case with respect to Lebesgue measure on the real line, but the ideas generalise to ℝ^k and other situations). Investigations therefore involve both analysis and number theory. It is the purpose of this paper to review the important contribution made by variants of the second Borel-Cantelli Lemma (also known as the divergence part of the Borel-Cantelli Lemma). While doing this we shall prove sharper variants than have previously appeared and give their applications.

The investigation of Pythagorian triples has a very long history. For the first hundred years I refer to the famous book [DIC01]. Triangles of this type were given by Greek and Indian mathematicians. Arithmetically these are the solutions of the diophantine equation in rational numbers. The general solution is given by the formulas .

Analytic number theory is about counting the number of sets of integers satisfying certain conditions. There are two famous questions.

A classical conjecture of E. Artin[Ar] predicts that any integer a ≠ ±1 or a perfect square is a primitive root (mod p) for infinitely many primes p. This conjecture is still open. In 1967, Hooley[H] proved the conjecture assuming the (as yet) unresolved generalized Riemann hypothesis for Dedekind zeta functions of certain number fields.

Let, as usual ℕ, ℤ, ℚ, ℝ, ℂ be the set of positive integers, integers, rational, real, and complex numbers, respectively. Let ℚ_×, ℝ_× be the multiplicative group of positive rationals, reals, respectively. Let \(\mathcal{P}\) be the set of prime numbers.

Ask any Ask any mathematician - indeed any number theorist - to state Hamburger’s theorem; chances are the response will be something like, “Riemann’s function ζ(s) is uniquely determined by its functional equation.” In fact, this is correct, as far as it goes, but (as is often the case) closer examination show that it does not go nearly far enough.

Our purpose is to give an overview of the applications of number theory to public-key cryptography. We conclude by describing some tantalizing unsolved problems of number theory that turn out to have a bearing on the security of certain cryptosystems.

The purpose of the present article is to survey some mean value results obtained recently in zeta-function theory. We do not mention other important aspects of the theory of zeta-functions, such as the distribution of zeros, value-distribution, and applications to number theory. Some of them are probably treated in the articles of Professor Apostol and Professor Ramachandra in the present volume.

Algebraic curves over finite fields with many rational points have received a lot of attention in recent years. We present a survey of this subject covering both the case of fixed genus and the asymptotic theory. A strong impetus in the asymptotic theory has come from a thorough exploitation of the method of infinite class field towers. On the other hand, we show by a counterexample that Perret’s conjecture on infinite class field towers is wrong, and so Perret’s method of infinite ramified class field towers breaks down. In the last two sections of the paper we discuss applications of algebraic curves over finite fields with many rational points to coding theory and to the construction of low-discrepancy sequences.

A classical question in the Theory of Numbers is one of expressing a positive integer as a sum of squares of integers. The qualitative aspects of this problem require at times no more than rudimentary congruence considerations e.g. a prime number leaving remainder 3 on division by 4 cannot be a sum of two squares of integers; however, in general, subtle arguments are called for. Fermat’s Principle of Descent needs to come into play for a proof of the (Euler-Fermat-) Lagrange theorem that every positive integer is a sum of four squares of integers. Skillful use of elliptic theta functions was made by Jacobi to obtain a quantitative refinement of that assertion, viz. according as n is an odd or even natural number, the number of ways of expressing n as a sum of four squares of integers is 8σ*(n) or 24σ*(n), where σ* (t) for any natural number t is the sum of all the odd natural numbers dividing t; Jacobi’s famous identity linking θ ₃ ⁴ with other theta constants θ₂, θ₄ and their derivatives is an analytic encapsulation of all these formulae for varying n. An analytic formulation of similar nature arises also as a special case of the Siegel formula (extended suitably to cover the boundary case of quaternary quadratic forms as well) which connects theta series associated with quadratic forms to Eisenstein series: for complex z with positive imaginary part, where the sum over p, q giving the Eisenstein series on the right hand side is only conditionally convergent and can be realized from an absolutely convergent Eisenstein series by analytic continuation via Hecke’s Grenzprozess (The inner sum over p is over all integers which are coprime to q and are of opposite parity to q).

Historically the theory of quadratic forms was regarded as a topic in number theory. However, Witt’s paper “Theorie der quadratischen Formen in beliebigen Körpern” of 1937[15] opened up a new chapter in the theory: that of combining the number theoretic aspect with the algebraic development, by the creation of the famous Witt ring.

We begin by stating the Prime Number Theorem in a way somewhat different from the usual. Let p _n denote the n-th prime (viz. p₁ = 2, p₂ = 3, p₃ = 5,…).

If x₁, x₂,…, x _s are integers positive negative or zero such that then (x₁, x₂,…, x _s ) is called a representation of n as a sum of s squares, and the total number of representations is denoted by R _s (n). Two representations (x₁, x₂,…, x _s ) and (y₁, y₂,…, y _s ) are considered to be different unless Further, using a notation introduced by J.W.L. Glaisher, we write R _{α, β} (n) for the number of representations of n as a sum of squares of which α are odd and β are even, no restriction being placed upon the order of the squares. Observe that R _s (0) = 1, and that R _{α, β} (0) = 1, if α = 0, but that otherwise R_{α, β}(0) = 0.

1⁰. There are two geometric disciplines which have the sufficient large intersection. Now we mean the discrete geometry and the geometry of numbers. It is accepted to refer various problems about dispositions of points and figures in a space to the discrete geometry. The same problems are accepted in the geometry of numbers, when they are somehow connected with point lattices in the n-dimensional euclidean space \({\mathbb{E}^n}\), and also some other problems about lattices are accepted as well.

This paper contains an account of the results on the following topics:

A survey on algebraic independence of transcendental numbers by Gel’fond’s method was published by the author in 1984 [84 W1]. Here we cover the recent period 1984/1997.