PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

For more information about this meeting, contact Robert Vaughan.

Title:The Skewes Number
Seminar:Algebra and Number Theory Seminar
Speaker:Roger Plymen, Southampton University
Abstract:
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
Date:10 / 11 / 2012
Time:11:15am - 12:05pm