More congruences for the coefficients of quotients of Eisenstein series
Let \(s\) be a complex number and let \(\sigma_s(n):=\sum_{d| n,d>0}d^s\) be the corresponding divisor function. Let \(B_n\) be the Bernoulli numbers and let \[ E_k(q):=1-\frac{2k}{B_k}\sum_{n=1}^\infty\sigma_{k-1}(n)q^n \] be the Eisenstein series, where \(k\) is a positive even integer, \(q=e^{2\pi i z}\), \(\Im z>0\). Let \(F_1(q):=\frac{1}{E_2(q)}\), \(F_2(q):=\frac{1}{E_4(q)}\), \(F_3(q):=\frac{1}{E_6(q)}\), \(F_4(q):=\frac{E_2(q)}{E_4(q)}\), \(F_5(q):=\frac{E_2(q)}{E_6(q)}\), \(F_6(q):=\frac{E_4(q)}{E_6(q)}\), \(F_7(q):=\frac{E_2(q)^2}{E_6(q)}\), with the Fourier expansions \[ F_i(q):=\sum_{n=0}^\infty a_i(n)q^n,\,i=1,\ldots, 7. \] The author proves: Theorem 1. If \(l\) is any positive integer, then the set of non-negative integers \(n\) satisfying \[ a_1(n)\equiv 0\pmod{3^l} \] has arithmetic density 1. If \(i\in\{2,4\}\), the prime \(p\) is equal to 3 or \(\equiv 5,11\pmod{12})\), and \(l\) is any positive integer, then the set of non-negative integers \(n\) satisfying \[ a_i(n)\equiv 0\pmod{p^l} \] has arithmetic density 1. If \(i\in\{3,5,6,7\}\), the prime \(p\) is equal to 3 or \(\equiv 7,11\pmod{12}\), and \(l\) is any positive integer, then the set of non-negative integers \(n\) satisfying \[ a_i(n)\equiv 0\pmod{p^l} \] has arithmetic density 1. Theorem 2. If \(1\leq i\leq 7\) and \(l\) is a positive integer, then the set of non-negative integers \(n\) satisfying \[ a_i(n)\equiv 0\pmod{2^l} \] has arithmetic density 1. Furthermore, there exists some integer \(w\) such that \[ a_i(mn')\equiv 0\pmod{2^l} \] whenever \(m\) is a square-free odd integer that is the product of at least \(w\cdot l\) odd prime factors, with \((m,n')=1.\)
scientific article; zbMATH DE number 5009839