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Mod p structure of alternating and non-alternating multiple harmonic sums

Jianqiang Zhao (2011)

Journal de Théorie des Nombres de Bordeaux

The well-known Wolstenholme’s Theorem says that for every prime p > 3 the ( p - 1 ) -st partial sum of the harmonic series is congruent to 0 modulo p 2 . If one replaces the harmonic series by ∑ k ≥ 1 1 / n k for k even, then the modulus has to be changed from p 2 to just p . One may consider generalizations of this to multiple harmonic sums (MHS) and alternating multiple harmonic sums (AMHS) which are partial sums of multiple zeta value series and the alternating Euler sums, respectively. A lot of results along this direction...

Modifications of the Eratosthenes sieve

Jerzy Browkin, Hui-Qin Cao (2014)

Colloquium Mathematicae

We discuss some cancellation algorithms such that the first non-cancelled number is a prime number p or a number of some specific type. We investigate which numbers in the interval (p,2p) are non-cancelled.

More on Divisibility Criteria for Selected Primes

Adam Naumowicz, Radosław Piliszek (2013)

Formalized Mathematics

This paper is a continuation of [19], where the divisibility criteria for initial prime numbers based on their representation in the decimal system were formalized. In the current paper we consider all primes up to 101 to demonstrate the method presented in [7].

Multidimensional Gauss reduction theory for conjugacy classes of SL ( n , ℤ )

Oleg Karpenkov (2013)

Journal de Théorie des Nombres de Bordeaux

In this paper we describe the set of conjugacy classes in the group SL ( n , ℤ ) . We expand geometric Gauss Reduction Theory that solves the problem for SL ( 2 , ℤ ) to the multidimensional case, where ς -reduced Hessenberg matrices play the role of reduced matrices. Further we find complete invariants of conjugacy classes in GL ( n , ℤ ) in terms of multidimensional Klein-Voronoi continued fractions.

Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions

Siao Hong, Shuangnian Hu, Shaofang Hong (2016)

Open Mathematics

Let f be an arithmetic function and S = {x1, …, xn} be a set of n distinct positive integers. By (f(xi, xj)) (resp. (f[xi, xj])) we denote the n × n matrix having f evaluated at the greatest common divisor (xi, xj) (resp. the least common multiple [xi, xj]) of x, and xj as its (i, j)-entry, respectively. The set S is said to be gcd closed if (xi, xj) ∈ S for 1 ≤ i, j ≤ n. In this paper, we give formulas for the determinants of the matrices (f(xi, xj)) and (f[xi, xj]) if S consists of multiple coprime...

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