# Meeting Details

Title: The Skewes Number Algebra and Number Theory Seminar Roger Plymen, Southampton University Let pi(x) denote the number of primes less than or equal to x, let li(x) denote the logarithmic integral \int_0^x dt/log t. The prime number theorem says pi(x)~li(x). The evidence from any table of primes suggests that pi(x) < li(x) for all x. Littlewood's theorem (1914) says that the diff erence pi(x)-li(x) changes sign infi nitely often. This implies that there is a least crossover, a least number X for which pi(X) > li(X). What is X? No-one knows. However, successive upper bounds have been given, starting with the famous bound due to Skewes: 10^10^10^43 We will bring the subject up to date with two papers from 2010: Chao-Plymen and Saouter-Demichel. The current world record is held by Stefanie Zegowitz: her upper bound is around exp(727.9513), a number with 316 digits. The statement is There exists an x < exp(727:9513) such that pi(x) > li(x). This work depends on the Riemann Explicit Formula { I'll begin with this formula.

### Room Reservation Information

Room Number: MB106 10 / 11 / 2012 11:15am - 12:05pm