Explicit Lower Bounds for Dirichlet Series of Higher Power Representation Functions

TL;DR

This paper develops explicit lower bounds for Dirichlet series associated with higher power representation functions, using geometric and analytic methods without relying on modular techniques.

Contribution

It introduces a novel analytic framework combining theta functions and cotangent series to derive bounds for higher power representation Dirichlet series.

Findings

Derived explicit lower bounds using geometric restrictions.

Applied Holder's inequality for alternative bounds.

Provided a flexible approach avoiding modular methods.

Abstract

We investigate Dirichlet-type series generated by representation functions that count the number of ways an integer can be expressed as a sum of 'k' signed higher even powers. By combining generalized theta generating functions with a family of generalized cotangent series introduced in previous work, we derive two distinct explicit lower bounds for these series. The first estimate arises from a geometric restriction of the lattice to its diagonal, while the second utilizes Holder's inequality on the integral representation of the series. The methods presented here avoid modular techniques and offer a flexible analytic framework for higher-power representation problems.