p-Integral harmonic sums

Integral Representations of Harmonic Lattice Sums

Quasi-symmetric functions and mod p multiple harmonic sums

We present a number of results about (finite) multiple harmonic sums modulo a prime, which provide interesting parallels to known results about multiple zeta values (i.e., infinite multiple harmonic series). In particular, we prove a “duality” result for mod p harmonic sums similar to (but distinct from) that for multiple zeta values. We also exploit the Hopf algebra structure of the quasi-symm...

Finiteness of p-Divisible Sets of Multiple Harmonic Sums∗

Abstract. Let l be a positive integer and s = (s1, . . . , sl) be a sequence of positive integers. In this paper we shall study the arithmetic properties of multiple harmonic sum H(s;n) which is the n-th partial sum of multiple zeta value series ζ(s). We conjecture that for every s and every prime p there are only finitely many p-integral partial sums H(s;n). This generalizes a conjecture of Es...

Mellin Transforms: Harmonic Sums

My assignment is going to introduce the Mellin transform and its application on harmonic sums [1]. Hjalmar Mellin(1854-1933, [2] for a summary of his works) gave his name to the Mellin transform, a close relative of the integral transforms of Laplace and Fourier. Mellin transform is useful to the asymptotic analysis of a large class of sums that arise in combinatorial mathematics, discrete prob...

Computing prime harmonic sums

We discuss a method for computing ∑ p≤x 1/p, using time about x2/3 and space about x1/3. It is based on the Meissel-Lehmer algorithm for computing the prime-counting function π(x), which was adapted and improved by Lagarias, Miller, and Odlyzko. We used this algorithm to determine the first point at which the prime harmonic sum first crosses 4.