Weierstrass Sigma Function Coefficients Divisibility Hypothesis

TL;DR

This paper investigates the divisibility properties of the coefficients in the series expansion of the Weierstrass sigma function, proposing a hypothesis that relates their 2-adic and 3-adic valuations to factorial expressions, with implications for the structure of the series.

Contribution

The paper introduces a divisibility hypothesis for the coefficients of the Weierstrass sigma function series, connecting their valuations to factorial-based formulas and suggesting a new algebraic structure.

Findings

Coefficients are integers with specific divisibility properties.

Proposed valuation formulas relate coefficients to factorial valuations.

If true, the sigma function forms a Hurwitz series over a particular ring.

Abstract

We consider the coefficients in the series expansion at zero of the Weierstrass sigma function \[ \sigma(z) = z \sum_{i, j \geqslant 0} {a_{i,j} \over (4 i + 6 j + 1)!} \left({g_2 z^4 \over 2}\right)^i \left(2 g_3 z^6\right)^j. \] We have $a_{i,j} \in \mathbb{Z}$. We present the divisibility Hypothesis for the integers $a_{i,j}$ \begin{align*} \nu_2(a_{i,j}) &= \nu_2((4i + 6j + 1)!) - \nu_2(i!) - \nu_2(j!) - 3 i - 4 j, & \nu_3(a_{i,j}) &= \nu_3((4i + 6j + 1)!) - \nu_3(i!) - \nu_3(j!) - i - j. \end{align*} If this conjecture holds, then $\sigma(z)$ is a Hurwitz series over the ring $\mathbb{Z}[{g_2 \over 2}, 6 g_3]$.