# Congruences modulo primes of the Romik sequence related to the Taylor expansion of the Jacobi theta constant {\theta}_3

## TL;DR

This paper investigates the congruence properties of a sequence derived from the Taylor expansion of the Jacobi theta constant, proving conjectures about its behavior modulo various primes, including specific residue classes.

## Contribution

It proves conjectures about the congruences of the Romik sequence related to the Jacobi theta constant, expanding understanding of its modular properties.

## Key findings

- d(n) ≡ (-1)^{n+1} (mod 5) for all n ≥ 1
- d(n) vanishes modulo p for large n when p ≡ 3 (mod 4)
- Some conjectured congruences are established for the sequence d(n)

## Abstract

Recently, Romik determined in [9] the Taylor expansion of the Jacobi theta constant \theta_3, around the point x = 1. He discovered a new integer sequence, (d(n))_0^\infty=1, 1, -1, 51, 849, -26199, \dots, from which the Taylor coefficients are built, and conjectured that the numbers d(n) satisfy certain congruences modulo various primes. In this paper, we prove some of these conjectures, for example that d(n)\equiv (-1)^{n+1}(mod 5) for all n\geq 1,and that for any prime p\equiv 3 (mod 4), d(n) vanishes modulo p for all large enough n.