| In this thesis, it is shown that a certain function of three complex variables has a domain of definition beyond the region obtained by simple estimates. Such an extension is known as an analytic continuation for a complex-valued function. The function studied is of particular interest because it contains an infinite family of Dirichlet L-functions. These are classical objects in analytic number theory originally introduced to study prime numbers and finer questions about prime numbers in arithmetic progressions. Over the last century, they have been greatly generalized to answer questions about solutions to Diophantine (i.e. integer-valued) equations. These rather complicated functions are able to be studied, in part, because they are amazingly symmetric about a line. One application of the analytic continuation of the original function of three variables is a “second-order” approximation (that is, a fine estimate known as the second moment) on the growth of these L-functions along the axis of symmetry.; The methods used exploit the symmetry of the L-functions. By determining the set of symmetries inherited by the function of three-variables, it can be shown that the function is naturally defined over a much larger region than the initial domain of definition. The function is an infinite series of weighted products of two varying Dirichlet L-functions. The values of the particular L-functions studied are multiples of cube roots of 1. Hence, we call our function a cubic multiple Dirichlet series. |
