From sums of divisors to partition congruences

Abstract

Let 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 ∑d|n(±1)d+1dz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{d|n} (\\pm 1)^{d+1}\\,d^z$$\\end{document}, ∑d|n(-1)n/d+1dz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{d|n} (-1)^{n/d+1}\\,d^z$$\\end{document} and ∑d|n(-1)n/d+ddz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{d|n} (-1)^{n/d+d}\\,d^z$$\\end{document}. 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.