Meromorphic Dirichlet Series

Abstract

This thesis studies several problems concerning the meromorphic continuation of Dirichlet series to the complex plane.

We show that if a Dirichlet series f(s) has a meromorphic continuation to the complex plane and the power series generating function of a sequence b(n) has a meromorphic continuation to z=1, then the Dirichlet series whose coefficients are the additive convolution of the coefficients of f(s) with the sequence b(n) has a meromorphic continuation to the complex plane. Using specific choices of the sequence b(n), we show that the Dirichlet series whose coefficients are a shift, forward or backward difference, or partial sum of the coefficients of f(s) has a meromorphic continuation to the complex plane. We study several examples of such Dirichlet series involving important arithmetic functions, including the Dirichlet series whose coefficients are the Chebyshev function, the Mertens function, the partial sums of the divisor function, or the partial sums of a Dirichlet character.

We apply these results to study the Dirichlet series whose coefficients are the sum of the base-b digits of the integers. We also study the Dirichlet series whose coefficients are the cumulative sum of the base-b digits of the integers less than n. We show that these Dirichlet series have a meromorphic continuation to the complex plane, and we give the locations of the poles and the residue at each pole. We also consider an interpolation of the sum-of-digits and cumulative sum-of-digits functions from integer bases b to a real parameter, and show that Dirichlet series attached to these interpolated sum-of-digits functions meromorphically continue one unit left of their halfplanes of convergence.

Finally, we consider a one-parameter family of Dirichlet series related to Ramanujan sums. The classical Ramanujan sum is a function of two integer variables; we replace one of the integer parameters with a complex number and consider the Dirichlet series attached to such complex Ramanujan sums. We show that this Dirichlet series continues to a meromorphic function of two complex variables and locate its singularities.