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 difference pi(x)-li(x) changes sign infinitely 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 |