A Fourier analysis approach to disprove the weak Shanks conjecture

TL;DR

This paper develops Fourier coefficient formulas for a specific function to generate counterexamples that disprove the weak Shanks conjecture, advancing understanding of its limitations.

Contribution

It introduces explicit formulas for Fourier coefficients of a complex function, enabling numerical optimization to find counterexamples to the weak Shanks conjecture.

Findings

Provided new counterexamples to the weak Shanks conjecture.

Derived formulas involving hypergeometric functions for Fourier coefficients.

Enabled numerical exploration of parameters to disprove the conjecture.

Abstract

We derive formulas for the Fourier coefficients of $|f|^2$, where $f(z_1,z_2)=(1-\frac{z_1+z_2}{r})^{-\alpha}$, in terms of hypergeometric functions. Using these formulas we provide additional counterexamples to the weak Shanks conjecture, which was recently disproven by B\'en\'eteau, Khavinson and Seco. The obtained formulas allow for (numerical) optimization over the parameters $\alpha$ and $r$.