Analysis Seminar

I will review some aspects of the potential theory on R^n with logarithmic kernels, and discuss the relation to several alternative approaches to estimating the prime counting function. The starting point of my inquiry is a short proof I will present of a (weak) form of the uncertainty principle on the real line, based on the obvious fact that the Riemann zeta function has (plenty of) zeros in the critical strip. The question of whether this type of connection could be worked into anything useful, either arithmetically or analytically, leads to a new outlook on the old Gelfond-Schnirelman topic. (The latter approach is notorious for falling short of the prime number theorem, as concluded by investigations circa 1980 by Bombieri, Chudnovsky, Nair, and others including Pritsker more recently, but nevertheless it led in the process to some unanswered questions about logarithmic potentials with external fields, and to a historic revival of the Selberg integral in the late 1970s.) Also in this light, I will introduce a related complex-analytic argument from a recent joint work with Calegari and Tang in transcendental number theory, and derive as a by-product yet another simple proof of a prime-counting lower bound stronger than Chebyshev's. We are led to formulate some problems in Fourier analysis motivated by questions from number theory