The Circle Problem of Gauss, the Divisor Problem of Dirichlet, and Ramanujan's Interest in Them

Abstract: Let $r_2(n)$ denote the number of representations of the positive integer $n$ as a sum of two squares, and let $d(n)$ denote the number of positive divisors of $n$. Gauss and Dirichlet were evidently the first mathematicians to derive asymptotic formulas for $\sum_{n\leq x}r_2(n)$ and $\sum_{n\leq x}d(n)$, respectively, as $x$ tends to infinity. The magnitudes of the error terms for the two asymptotic expansions are unknown. Determining the exact orders of the error terms are the Gauss Circle Problem and Dirichlet's Divisor Problem, respectively, and they represent two of the most famous and difficult unsolved problems in number theory.

Beginning with his first letter to Hardy, it is evident that Ramanujan had a keen interest in the Divisor Problem, and from a paper written by Hardy and published in 1915, shortly after Ramanujan arrived in England, we learn that Ramanujan was also greatly interested in the Circle Problem. In a fragment published with his Lost Notebook, Ramanujan stated two doubly infinite series identities involving Bessel functions that we think Ramanujan derived to attack these two famous unsolved problems. The identities are difficult to prove. Unfortunately, we cannot figure out how Ramanujan might have intended to use them. We survey what is known about these two unsolved problems, with a concentration on Ramanujan's two marvelous and mysterious identities. Joint work with Sun Kim, Junxian Li, and Alexandru Zaharescu is discussed.

number theory

Audience: researchers in the topic

EIMI Number Theory Seminar

Series comments: Password: the number of quadratic nonresidues modulo 23