Congruences like Ramanujan's for powers of the partition function
Let \(p(n)\) be the partition function. The congruences of Ramanujan mentioned in the title of this paper are the three congruences \(p(5n+4)\equiv 0\pmod 5\), \(p(7n+5)\equiv 0\pmod 7\), \(p(11n+6)\equiv 0 \pmod {11}\), each being true for all nonnegative integers \(n\). For each integer \(r\), the function \(p_ r(n)\) is defined by means of the equation \(P(q)^ r=\sum_{n=0}^ \infty p_ r(n)q^ n\), where \(P(q)=\sum_{n=0}^ \infty p(n)q^ n=\prod_{n=1}^ \infty (1-q^ n)^{-1}\) is the generating function for \(p(n)\). Then \(p_ r(n)\) is an integer for all integers \(r\) and positive integers \(n\). If \(\pi\) is a prime \(\geq 5\), \(r\) is an integer, and \(a\in[0,1,\dots,\pi- 1]\), then it is said that there is a congruence for \(p\) at \((\pi,r,a)\) if \(p_ r(n\pi+a)\equiv 0 \pmod \pi\) for all nonnegative integers \(n\). A congruence for \(p\) at \((\pi,r,a)\) is called exceptional if \(r\not\in[\pi- 1,\pi-3]\). The theory of reductions of modular forms modulo \(\pi\) is used to prove that if there is an exceptional congruence for \(p\) at \((\pi,r,a)\) then \(r\) is odd and \(24a\equiv r \pmod \pi\). It is then shown that there is a congruence for \(p\) at \((\pi,\pi-1,a)\) if and only if \(24a+1\) is a quadratic nonresidue modulo \(\pi\), and there is a congruence for \(p\) at \((\pi,\pi-3,a)\) if and only if either \(8a+1\equiv 0 \pmod \pi\) or \(8a+1\) is a quadratic nonresidue modulo \(\pi\). The final theorem of this paper states that, for all nonnegative integers \(n\), \[ \begin{alignedat}{3} p_ 3(11n+7)&\equiv 0\pmod {11},&\quad p_ 5(11n+8)&\equiv 0\pmod {11},&\quad p_ 7(11n+9)&\equiv 0\pmod {11},\\ p_ 3(17n+15)&\equiv 0\pmod {17},&\quad p_ 9(17n+11)&\equiv 0\pmod {17},&\quad p_{13}(17n+14)&\equiv 0\pmod {17},\\ p_ 7(19n+9)&\equiv 0\pmod {19}, &\quad p_ 9(19n+17)&\equiv 0\pmod {19}, &\quad p_{13}(19n+14)&\equiv 0\pmod {19}.\end{alignedat} \] {}.
scientific article; zbMATH DE number 119864