Lior Rosenzweig: Prime polynomials in short intervals

There is an analogy between Z, the ring of integers, and F_q[t], the ring of polynomials over a finite field with q elements. We will demonstrate this analogy through the following problem about prime numbers in short intervals.

By the Prime Number Theorem, the distribution of primes around a number x, is 1/log(x). This suggests that in an interval near x, say (x,x+y], the expected number of primes is y/log(x). The Prime Number Theorem itself shows that this is true for values of y almost as large as x, and by the Riemann Hypothesis one can take y as small as x^{1/2+epsilon}. Conjecturally we can take y as small as x^epsilon.

The talk is based on a joint work with Efrat Bank and Lior Bary-Soroker