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Explicit Lower Bounds for Dirichlet Series of Higher Power Representation Functions

Mahipal Gurram

TL;DR

The paper studies the Dirichlet-type series S_{m,k}(a) associated with representing integers as sums of k even powers and derives two explicit, modular-free lower bounds: one from restricting to the diagonal lattice (geometric) and another from Hölder's inequality on an integral representation (analytic). Both bounds are expressed through the generalized cotangent series U_{2m} and generalized theta functions Θ_m, and are valid for k<2m. The authors illustrate the bounds in cases m=1 and m=2, revealing regime-dependent dominance: geometry prevails for small a while convexity-based analysis dominates for large a. Overall, the work provides a flexible analytic framework for higher-power representation problems and clarifies the trade-off between lattice geometry and convexity across dimensions.

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.

Explicit Lower Bounds for Dirichlet Series of Higher Power Representation Functions

TL;DR

The paper studies the Dirichlet-type series S_{m,k}(a) associated with representing integers as sums of k even powers and derives two explicit, modular-free lower bounds: one from restricting to the diagonal lattice (geometric) and another from Hölder's inequality on an integral representation (analytic). Both bounds are expressed through the generalized cotangent series U_{2m} and generalized theta functions Θ_m, and are valid for k<2m. The authors illustrate the bounds in cases m=1 and m=2, revealing regime-dependent dominance: geometry prevails for small a while convexity-based analysis dominates for large a. Overall, the work provides a flexible analytic framework for higher-power representation problems and clarifies the trade-off between lattice geometry and convexity across dimensions.

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.

Paper Structure

This paper contains 10 sections, 2 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. The Geometric Lower Bound
  3. The Analytic Lower Bound
  4. Illustrative Examples
  5. Case $m=1$: Sums of Squares
  6. Case $m=2$: Sums of Fourth Powers
  7. Detailed Comparative Analysis
  8. Dependence on the Spectral Parameter $a$
  9. Dimensional Stability vs. Convexity
  10. Conclusion

Key Result

Theorem 2.1

Let $m,k\in\mathbb{N}$ such that $k < 2m$, and let $a>0$. Let $U_{2m}(z)$ be the generalized cotangent series defined by Then the Dirichlet series associated with $r_{m,k}(n)$ satisfies

Theorems & Definitions (5)

  • Theorem 2.1: Geometric Lower Bound
  • proof
  • Theorem 3.1: Analytic Lower Bound
  • proof
  • Remark 5.1: Regime Transition