The Congruence Properties of Romik’s Sequence of Taylor Coefficients of Jacobi’s Theta Function $θ_3$

Abstract

In [Ramanujan J. 52 (2020), 275-290], Romik considered the Taylor expansion of Jacobi's theta function $\theta_3(q)$ at $q=e^{-\pi}$ and encoded it in an integer sequence $(d(n))_{n\ge0}$ for which he provided a recursive procedure to compute the terms of the sequence. He observed intriguing behaviour of $d(n)$ modulo primes and prime powers. Here we prove (1) that $d(n)$ eventually vanishes modulo any prime power $p^e$ with $p\equiv3$ (mod 4), (2) that $d(n)$ is eventually periodic modulo any prime power $p^e$ with $p\equiv1$ (mod 4), and (3) that $d(n)$ is purely periodic modulo any 2-power $2^e$. Our results also provide more detailed information on period length, respectively from when on the sequence vanishes or becomes periodic. The corresponding bounds may not be optimal though, as computer data suggest. Our approach shows that the above congruence properties hold at a much finer, polynomial level.