A394831 Absolute norms of finite mock theta sums T_p(zeta_p), where p runs through the odd primes.

3, 1, 337, 1871, 1945789, 122489047, 4009179817, 23886447238763, 1344308567205093949, 347053586255000098783, 52082013528217314295322509, 20137281069484387254399485149, 213584407983023063027316287926819, 69612789356186134219934136566519267

Offset

1,1

Comments

Let p be the n-th odd prime, so p = prime(n+1), and let zeta_p = exp(2*Pi*I/p). Define T_p(z) = Sum_{m=0..p-1} z^(m^2) / (-z;z)_m^2, where (-z;z)_m = Product_{j=1..m} (1 + z^j). Then a(n) is the absolute norm of T_p(zeta_p) from Q(zeta_p) to Q.

The terms were computed by exact arithmetic in Q(zeta_p), not by numerical approximation.

For odd prime p, each factor 1 + zeta_p^j, 1 <= j <= p-1, is a cyclotomic unit, since Norm(1 + zeta_p^j) = Phi_p(-1) = 1. Therefore T_p(zeta_p) is an algebraic integer, and its norm is an integer.

Empirical observation: for 1 <= n <= 14, every rational prime ell dividing a(n), with ell != p, satisfies ell == 1 mod 2*p.

Equivalently, the observed prime divisors ell != p split completely in Q(zeta_{2p}).

References

K. Bringmann and L. Rolen, Radial limits of mock theta functions, Res. Math. Sci. 2:17 (2015).

A. Folsom, K. Ono, and R. C. Rhoades, Mock theta functions and quantum modular forms, Forum Math. Pi 1 (2013), e2.

Links

Table of n, a(n) for n=1..14.

Formula

Let p = prime(n+1), and let zeta_p = exp(2*Pi*I/p). Then a(n) = abs(Norm_{Q(zeta_p)/Q}(T_p(zeta_p))), where T_p(z) = Sum_{m=0..p-1} z^(m^2) / Product_{j=1..m} (1 + z^j)^2.

Equivalently, let U_p(x) be the unique polynomial of degree < p-1 representing T_p(x) modulo Phi_p(x), where the denominators are inverted modulo Phi_p(x). Then a(n) = abs(Resultant(Phi_p(x), U_p(x))).

Example

For n = 1, p = 3. We have T_3(zeta_3) = 2 + zeta_3, so a(1) = abs(Norm(2 + zeta_3)) = 3.
For n = 2, p = 5. We have T_5(zeta_5) = 2 + zeta_5 + zeta_5^2 + 2*zeta_5^3, whose norm is -1, so a(2) = 1.
For n = 3, p = 7. We have T_7(zeta_7) = 3 + zeta_7 + zeta_7^3 + zeta_7^4 + 2*zeta_7^5, whose norm has absolute value 337, so a(3) = 337.

Prog

(SageMath)
from sage.all import *
def odd_prime(n):
    p = ZZ(3)
    for _ in range(n - 1):
        p = next_prime(p)
    return p
def T_norm(p):
    K = CyclotomicField(p)
    z = K.gen()
    T = K(0)
    poch = K(1)
    for m in range(p):
        if m > 0:
            poch *= 1 + z^m
        T += z^(m*m) / poch^2
    return ZZ(abs(T.norm()))
def a(n):
    return T_norm(odd_prime(n))
print([a(n) for n in range(1, 15)])

Crossrefs

Cf. A000040, A393567.

Sequence in context: A367948 A369187 A266363 * A382849 A068542 A382793

Adjacent sequences: A394828 A394829 A394830 * A394832 A394833 A394834

Keyword

nonn

Author

Joesph Daniel Burke III, May 02 2026

Status

approved