## Zeros of Dirichlet L-series on the critical line

### Volume 93 / 2000

#### Abstract

Introduction. In 1974, N. Levinson showed that at least 1/3 of the zeros of the Riemann ζ-function are on the critical line ([19]). Today it is known (Conrey, [6]) that at least 40.77% of the zeros of ζ(s) are on the critical line and at least 40.1% are on the critical line and are simple. In [16] and [17], Hilano showed that Levinson's original result is also valid for Dirichlet L-series. This paper is a shortened version of parts of the dissertation [3], the full details of which may be found at http://www.math.uni-frankfurt.de/~pbauer/diss.ps. We shall prove a mean value theorem for Dirichlet L-series and use this for proving some corollaries concerning the distribution of the zeros of L-series - amongst other results we improve the above mentioned bounds for Dirichlet L-series.