Congruence classes for modular forms over small sets. (English) Zbl 07872252
Summary: Serre showed that for any integer \(m\), \(a(n) \equiv 0 \pmod m\) for almost all \(n\), where \(a(n)\) is the \(n\)th Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study \(\# \{a(n) \pmod m\}_{n \leq x}\) over the set of integers with \(O(1)\) many prime factors. Moreover, we show that any residue class \(a \in \mathbb{Z}/m \mathbb{Z}\) can be written as the sum of at most 13 Fourier coefficients, which are polynomially bounded as a function of \(m\).
MSC:
| 11F30 | Fourier coefficients of automorphic forms |
| 11L07 | Estimates on exponential sums |
| 11P05 | Waring’s problem and variants |
| 11B37 | Recurrences |
| 11F80 | Galois representations |
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