##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Keith Conrad

#### UCSD

## The Mobius function in characteristic p

##### Abstract:

For an irreducible polynomial $f(T)$ in ${\\bf Z}[T]$ whose values are not all multiples of a common prime, the sequence $\\mu(f(n))$ is not expected to have any periodicity properties. In contrast, there can be periodicity when $f(T) \\in {\\bf F}[u][T]$ with $\\bf F$ a finite field. That is, the sequence $\\mu(f(g))$ can be periodic as $g$ runs over ${\\bf F}[u]$. This is based on peculiarities of characteristic p. We will briefly discuss the case of odd characteristic, and then focus on the extra subtleties of characteristic 2, where we make an interesting application of the residue theorem for a certain rational differential form. The ideas will be made explicit by treating a concrete case: as $g$ runs over ${\\bf F}_2[u]$, $\\mu(g^8+(u^3+u)g^4+u) = 1$. Remark: Note the unusual day, time, and room for the number theory seminar this week.

Host:

### June 6, 2003

### 12:30 PM

### AP&M 6438

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