363 research outputs found
Simple zeros of primitive Dirichlet -functions and the asymptotic large sieve
Assuming the Generalized Riemann Hypothesis (GRH), we show using the
asymptotic large sieve that 91% of the zeros of primitive Dirichlet
-functions are simple. This improves on earlier work of \"{O}zl\"{u}k which
gives a proportion of at most 86%. We further compute an -analogue of the
Pair Correlation Function averaged over all primitive Dirichlet
-functions in the range . Previously such a result was
available only when the average included all the characters .Comment: This work was initiated during the Arithmetic Statistics MRC program
at Snowbird, Utah. Corollary 3 and Section 7 are adde
The mean square of the product of a Dirichlet -function and a Dirichlet polynomial
We establish an analogue of a conjecture of Balasubramanian, Conrey, and
Heath-Brown for the family of all Dirichlet characters with conductor up to
. This forms another application of our work in developing an asymptotic
large sieve.Comment: 28 page
Mean values with cubic characters
We investigate various mean value problems involving order three primitive
Dirichlet characters. In particular, we obtain an asymptotic formula for the
first moment of central values of the Dirichlet L-functions associated to this
family, with a power saving in the error term. We also obtain a large-sieve
type result for order three (and six) Dirichlet characters.Comment: 22 pages; greatly shortened, simplified and corrected versio
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