A theorem in asymptotic number theory

An asymptotic formula in additive number theory

1 . Introduction . In his paper [1], Erdös introduced the sequences of positive integers b 1 < b, < . . ., with (b ;, bj ) = 1, for i ~A j, and 'bi 1 < oo . With any such arbitrary sequence of integers, he associated the sequence {di} of all positive integers not divisible by any bj , and he showed that if b1 > 2, there exists a 0 < 1 (independent of the sequence {b i }) such that d i 1 di < d°...

Asymptotic Group Theory, Volume 48, Number 4

W hat is the probability that two randomly chosen permutations of 1, . . . , n generate the group Sn of all permutations of 1, . . . , n ? How many subgroups of index n can a group with two generators have? What can be said about a group of order 2n which has a nonidentity commutator of length n− 10? These questions all concern different aspects of a field which is very active nowadays and whic...

Generalizations of a Classical Theorem in Number Theory

A classical theorem conjectured by Jacobi asserts that for an odd prime p, the sum of the quadratic residues in the interval (0, p) is less than the sum of the quadratic nonresidues if and only if p ■ 3 (mod 4). We generalize Jacobi's problem to fcth powers (mod p), k > 2, and we consider in some detail a generalization of Jacobi's conjecture to quadratic residues and nonresidues (mod n), n an ...

Analytic Number Theory and Dirichlet’s Theorem

In this paper, we prove Dirichlet’s theorem that, given any pair h, k with (h, k) = 1, there are infinitely many prime numbers congruent to h (mod k). Although this theorem lies strictly within the realm of number theory, its proof employs a range of tools from other branches of mathematics, most notably characters from group theory and holomorphic functions from complex analysis.

Two classic theorems from number theory: The Prime Number Theorem and Dirichlet’s Theorem