Arithmetic formulas for the Fourier coefficients of Hauptmoduln of level 2, 3, and 5

Abstract

We give arithmetic formulas for the coefficients of Hauptmoduln of higher levels as analogues of Kaneko's formula for the elliptic modular j-invariant. We also obtain their asymptotic formulas by employing Murty-Sampath's method.

Sign Changes of Coefficients of Powers of the Infinite Borwein Product

“…(1.30) When (t, m) ∈ {(2, 24), (3,12), (5,6)}, by a direct calculation of α (m) t (n) and Corollary 1.2, we recover (1.9)- (1.11).…”

“…See [13] for explicit definition of t(d). Matsusaka and Osanai [12] found similar formulas for c (m) t (n) with (t, m) ∈ {(2, 24), (3,12), (5,6)}. Based on these formulas and Laplace's method, they [12] proved that as n → ∞, c n ≡ 4 (mod 5).…”

“…For any prime p, thanks to the works of Andrews [3] and Schlosser and Zhou [15], we now know that {c p (n)} are both weak UPS sequences with period p. In fact, later we will show that they are actually UPS sequences with p as the least period of sign. By the work of Matsusaka and Osanai [12], from (1.9)-(1.11) we know that {c (3,12), (5,6)}. We should be aware that not all coefficients generated by infinite products are (weak) UPS sequences.…”

“…Based on these formulas and Laplace's method, they [12] proved that as n → ∞, c n ≡ 4 (mod 5). (1.11) This means that c (24) 2 (n), c (12) 3 (n) and c (6) 5 (n) possesses a sign-change property for large n. Later Hu and Ye [10] proved that for any n ≥ 0, (−1) n c (24) 2 (n) > 0.…”

The Fourier coefficients of the McKay–Thompson series and the traces of CM values

“…Ohta [12] and the author and Osanai [11] obtained the analogues of Kaneko's formula (1.2) in the cases of N = p = 2, 3, and 5, first found experimentally, and then showed the coincidence of q-series by using the Riemann-Roch theorem. In the present paper, we use the theory of Jacobi forms to generalize (1.2).…”