Convergence formulae for double Dirichlet series

Abstract

Regular convergence of double Dirichlet series was originally studied by Kojima (Rep Tohoku Univ 9:351–400, 1920), where the author proved a formula for what he called the related abscissae of regular convergence, allowing him to describe the sets of regular convergence of such double series in an extensive study. In this work we give a shorter and more comprehensive proof for his main result, a formula for the related abscissae. We also give two new formulae which are easier to work with and which extend the traditional formulae of convergence for the abscissa of an ordinary Dirichlet series to the double case. These new formulae can be adapted for the general double ones with some particular frequencies which grow at a pace very similar to the pace at which the sequence $$\{\log n\}_n$$ grows. Finally, we apply these new formulae to give examples of double Dirichlet series whose sets of regular convergence are not trivially obtained from the one variable case.