For more information about this meeting, contact Robert Vaughan.

Title: | Mean Values of Multiplicative Functions |

Seminar: | Algebra and Number Theory Seminar |

Speaker: | Robert Vaughan, The Pennsylvania State University |

Abstract: |

In 1968 Halasz showed that any multiplicative function f for which |f(n)| <= 1 and the associated Dirichlet series F(s)= \sum_{n=1}^{\infty} f(n)n^{-s} satisfies F(s)=o(1/(\sigma-1)) as \sigma tends to 1 from above must also satisfy S_0(x) = \sum_{n\le x} f(n) = o(x) as x\rightarrow\infty. Later he gave a quantitative version of this. One consequence is that it is an immediate consequence of \zeta(1+it) being non-zero that M(x)=\sum_{n\le x}\mu(n)=o(x) fron which the prime number theorem follows elementarily. In joint work with Hugh Montgomery we obtain a similar conclusion for the much more delicate situation regarding the behaviour of S_1(x) = \sum_{n\le x} f(n)/n in terms of that of F(s) as \sigma tends to 1 from above. There are numerous applications. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 10 / 04 / 2007 |

Time: | 11:15am - 12:05pm |