A Numerical Study of the Division Table

Abstract. An elementary division table, followed honestly, arrives at the summatory divisor function, the harmonic numbers, and — after one column of its 0-1 shadow is falsified — the Mertens function, so that det R_n = O(n^(1/2+e)) is the Riemann hypothesis wearing a matrix. A walk across the table traces an arch which is Fermat's factorisation method; its failure is repaired by Kraitchik and the quadratic sieve; rotating the picture forty-five degrees lands on the Dirichlet divisor and Gauss circle problems and Voronoi's series. Two closing sections truncate the sheet instead of walking it, reaching Buchstab's function and the wheel sieve. No result in the paper is new; its ledger lists twenty-one items and decomposes all twenty-one.