Hilbert gave an example of a bounded linear operator from l^2 to l^2 whose row and column sums are divergent. The result is now known as Hilbert's inequality. In 1911, Schur determined the optimal constant in Hilbert's inequality to be pi.

In 1974, Montgomery and Vaughan proved a weighted generalization of Hilbert's inequality and used it to estimate mean values of Dirichlet series. The optimal constant C in this generalized Hilbert inequality is known to be in the interval [pi, pi+1). It is widely conjectured that C=pi.

In this talk, I will consider the only known approach to obtain an upper bound for C and show that it cannot achieve C=pi. If time permits, I will discuss applications of the generalized Hilbert inequality, including the large sieve and the mean square estimate of the remainder term in the Dirichlet divisor problem. Speaker(s): Wijit Yangjit (University of Michigan)