Congruences and conjectures for the partition function (Q2782402)

A new construction of half-integral weight modular forms whose coefficients capture values of \(p(n)\) modulo \(l\), for primes \(l\geq 5\), is given. Explicit examples are given in the cases \(l=5,7\) and 11. For example, NEWLINE\[NEWLINE\sum_{n=0}^\infty p(5n+1) q^{5n+1}+ \sum_{n=0}^\infty p(5n+2) q^{5n+2}\equiv q(q;q)_\infty^{23} \pmod 5,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\begin{multlined} \sum_{n=0}^\infty p(7n+1) q^{7n+1}+ \sum_{n=0}^\infty p(7n+3) q^{7n+3}+ \sum_{n=0}^\infty p(7n+4) q^{7n+4}\\ \equiv q(q;q)_\infty^{23} \biggl( 1+240 \sum_{j=1}^\infty \frac{j^3q^j} {1-q^j} \biggr)^3+ 3q^2 (q;q)_\infty^{47} \pmod 7. \end{multlined}NEWLINE\]NEWLINE The modulo 11 result is longer. NEWLINENEWLINENEWLINEThe paper ends with a list of six general conjectures, a problem and a speculation. Some supporting numerical evidence is given.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00024].