Congruence properties of coefficients of the eighth-order mock theta function $$V_0(q)$$

Abstract

In [Ramanujan J. 52 (2020), 275-290], Romik considered the Taylor expansion of Jacobi's theta function θ 3 (q) at q = e −π and encoded it in an integer sequence (d(n)) n≥0 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 ≡ 3 (mod 4), (2) that d(n) is eventually periodic modulo any prime power p e with p ≡ 1 (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. 2020 Mathematics Subject Classification. Primary 11F37; Secondary 11B83 14K25. Key words and phrases. Modular forms of half integer weight, Jacobi theta function, Taylor coefficients, congruences.