THE p-ZASSENHAUS FILTRATION OF A FREE PROFINITE GROUP AND SHUFFLE RELATIONS

For a prime number p and a free profinite group S on the basis X, let $S_{\left (n,p\right )}$ , $n=1,2,\dotsc ,$ be the p-Zassenhaus filtration of S. For $p>n$ , we give a word-combinatorial description of the cohomology group $H^2\left (S/S_{\left (n,p\right )},\mathbb {Z}/p\right )$ in terms of the shuffle algebra on X. We give a natural linear basis for this cohomology group, which is constructed by means of unitriangular representations arising from Lyndon words.

Identities for Dirichlet and Lambert-type series arising from the numbers of a certain special word

The goal of this paper is to give several new Dirichlet-type series associated with the Riemann zeta function, the polylogarithm function, and also the numbers of necklaces and Lyndon words. By applying Dirichlet convolution formula to number-theoretic functions related to these series, various novel identities and relations are derived. Moreover, some new formulas related to Bernoulli-type numbers and polynomials obtain from generating functions and these Dirichlet-type series. Finally, several relations among the Fourier expansion of Eisenstein series, the Lambert series and the number-theoretic functions are given.