From sums of divisors to partition congruences

Abstract

AbstractLet z be a complex number. For any positive integer n it is well known that the sum of the zth powers of the positive divisors of n can be computed without knowing all the divisors of n, if we take into account the factorization of n. In this paper, we rely on the integer partitions of n in order to investigate computational methods for $$\sum _{d|n} (\pm 1)^{d+1}\,d^z$$ ∑ d | n ( ± 1 ) d + 1 d z , $$\sum _{d|n} (-1)^{n/d+1}\,d^z$$ ∑ d | n ( - 1 ) n / d + 1 d z and $$\sum _{d|n} (-1)^{n/d+d}\,d^z$$ ∑ d | n ( - 1 ) n / d + d d z . To compute these sums of divisors of n, it is sufficient to know the multiplicity of 1 in each partition involved in the computational process. Our methods do not require knowing the divisors of n or the factorization of n. New congruences involving Euler’s partition function p(n) are experimentally discovered in this context.