Résumé:

The title refers to the exact lower bound on the L1 norms, on the unit circle, of exponential sums with a given number of terms N. In 1979, this bound was proven to be on the order of C log(N) by Konyagin and, independently, by McGehee, Pigno and Smith. The exact lower bound was conjectured by Hardy and Littlewood to be equal to precisely the L1 norm of the geometric series with N terms. (For N odd, these are just the L1 norms of Dirichlet kernels; also known as the Lebesgue constants). This conjecture is still open. We will discuss a "new and promising" method, including a special case of the conjecture which this method yields immediately.