Seminar

Exponential sums over Z or Z^d are basic objects in Analytic Number Theory and oscillatory integrals over R or R^d are basic objects in Harmonic Analysis. These objects are quite different; for oscillatory integrals over R, a single continuum of scales is often sufficient for the analysis whereas for exponential sums over Z, every prime p gives rise to a family of scales {p^k}, all needed in the analysis. Nevertheless if one fixes the prime p and carries out the analysis at the corresponding scales (e.g.  by examining exponential sums over Z/p^k Z,  k=1,2,3,...) then the analogy to oscillatory integrals in euclidean settings is uncanny.

We will illustrate this in the simple setting of polynomial congruences and formulate some problems in elementary number theory in a way that harmonic analysts can appreciate and be able to use their prior acquired intuition.