The Laplace transform of Dirichlet L-functions

Laplace transform of certain functions with applications

The Laplace transform of the functions tν(1+ t)β, Reν > −1, is expressed in terms of Whittaker functions. This expression is exploited to evaluate infinite integrals involving products of Bessel functions, powers, exponentials, and Whittaker functions. Some special cases of the result are discussed. It is also demonstrated that the famous identity ∫∞ 0 sin(ax)/xdx =π/2 is a special case of our ...

Calculation of Dirichlet L-Functions

A method for calculating Dirichlet L-series is presented along with the theory of residue class characters and their automatic generation. Tables are given of zeros of Lseries for moduli S 24.

On the Order of Dirichlet L-functions

The Mean-square of Dirichlet L-functions

where α and β are small complex numbers satisfying α, β ≪ 1/ log q. Ingham [Ing] considered an analogous moment for the Riemann zeta-function on the critical line with small shifts. Paley [Pal] considered the moment above for Dirichlet L-functions. Heath-Brown [HB] has computed a similar moment, but for all characters modulo q, in the case that α = β = 0. His result is Theorem 1 (HB). There are...

The Fourth Moment of Dirichlet L-functions

Here ∑∗ denotes summation over primitive characters χ (mod q), φ(q) denotes the number of primitive characters (mod q), and ω(q) denotes the number of distinct prime factors of q. Note that φ(q) is a multiplicative function given by φ(p) = p − 2 for primes p, and φ(p) = p(1 − 1/p) for k ≥ 2 (see Lemma 1 below). Also note that when q ≡ 2 (mod 4) there are no primitive characters (mod q), and so ...