For more information about this meeting, contact Robert Vaughan.
| Title: | The Skewes Number |
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| Seminar: | Algebra and Number Theory Seminar |
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| Speaker: | Roger Plymen, Southampton University |
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| Abstract: |
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| 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 |
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| Date: | 10 / 11 / 2012 |
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| Time: | 11:15am - 12:05pm |
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