2025/09/24 by Ono, Ken
#05A30 #11F03 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2509.20316
In 1975, G. E. Andrews challenged the mathematics community to address L. Ehrenpreis' problem, which was to directly prove the modularity of the Rogers-Ramanujan q-series' summatory forms. This question is important because many different q-series appearing in combinatorics, representation theory, and physics often seem to be mysteriously modular, yet there is no general test to confirm this directly from the exotic q-series expressions. In this note, we answer the challenge. We use q-series algebra, first-order q-differential systems, and analytic continuation with monodromy to give a criterion that decides when such series are modular. Specifically, we establish a necessary and sufficient condition for a vector of holomorphic q-series on |q|<1 to form a vector-valued modular function without modular input, providing a clear path to modularity for strange q-series.