Short addition sequences for theta functions

TL;DR

This paper introduces efficient addition sequences for computing q-series in modular forms and elliptic functions, significantly reducing multiplications needed and improving computational speed.

Contribution

It presents novel short addition sequences for q-series, leveraging quadratic progressions, and a baby-step giant-step algorithm that outperforms previous methods.

Findings

Reduction to N + o(N) multiplications for sum computation

Representation of quadratic progressions as sums of smaller ones

Algorithm achieves O(N/ log r N) multiplications, beating previous lower bounds

Abstract

The main step in numerical evaluation of classical Sl2 (Z) modular forms and elliptic functions is to compute the sum of the first N nonzero terms in the sparse q-series belonging to the Dedekind eta function or the Jacobi theta constants. We construct short addition sequences to perform this task using N + o(N) multiplications. Our constructions rely on the representability of specific quadratic progressions of integers as sums of smaller numbers of the same kind. For example, we show that every generalised pentagonal number c 5 can be written as c = 2a + b where a, b are smaller generalised pentagonal numbers. We also give a baby-step giant-step algorithm that uses O(N/ log r N) multiplications for any r > 0, beating the lower bound of N multiplications required when computing the terms explicitly. These results lead to speed-ups in practice.