ABSTRACT
It was Euler who gave us the idea of a generating function. He was interested in the theory of partitions of a positive integer n. We denote by p(n), the number of partitions of n. Through the introduction of generating functions and the progress in the theory of functions of a complex variable, the study of partitions became rigorous. The functions related to partitions and formulae connected with them were discovered as special cases of a more general set-up involving theta functions and modular functions. They were investigated thoroughly by Carl Gustav Jacob Jacobi and others. The results found a place in additive number theory legitimately. The role of generating functions in additive number theory is similar to the role of Dirichlet series of arithmetic functions in multiplicative number theory. In 1859, B. Riemann (1826–1866) undertook the study of pi(x), the number of primes less than or equal to x in establishing Gauss’s conjecture pi(x)∼ xlog x and connected this problem with the properties of ζ(s) = ∑∞n=1 n−s(Re s > 1). Indeed, Riemann was one of the founders of the theory of functions of a complex variable and it was his interest in pi(x) that prompted him to pursue the general theory of functions of a complex variable. The Dirichlet series of an arithmetic function generalizes the Riemann ζ-function. In fact, ζ(s) is the generating function of the function e, where e(n) = 1, n≥ 1. The inverse of ζ(s) gives the Möbius function µ, where 1/ζ(s) =∑∞n=1µ(n)n−s, Re s> 1.
