Inspired by investigations of zeta functions, and old problem of Ankeny and Chowla asks whether any cosine polynomial f_A(x)=cos(a_1 x)+ ... +cos(a_n x), for an arbitrary set A={a_1,...a_n} of n distinct positive integers, must take a large negative value for some x in [0,2 pi]. Chowla later conjectured that the largest negative value of f_A is always at least of order n^1/2^, for any set A of size n. A refinement of Bourgain's approach due to Ruzsa gave the previous record bound of exp(sqrt(log n)). In this talk, we discuss recent progress establishing the first polynomial bound n^c^ with exponent c=1/7. We remark that Jin, Milojevic, Tomon and Zhang independently proved a polynomial bound with exponent c approximately 1/100 using a different method.
