Researches On The Related Problems Of Multiple Alternating Zeta Functions And Harmonic Numbers

The research on various forms of multiple zeta functions is important to general zeta function theory,airthmetic geometry,quantum groups,etc.In this paper,we mainly study the related property of alternating multiple zeta func-tions,and obtain some new identities.Meanwhile,considering the extension of the correlation identities between the classical harmonic series and the gen-eralized binomial coefficients to the m-order generalized harmonic series,the expression of the correlation sum is obtained.Based on the divisibility prop-erties of various forms of multiple harmonic sums,some congruences involving the sum of products of harmonic numbers and binomial coefficients have been obtained.The main contents are as follows:1.Using the correlation properties of the Bernoulli polynomials and the harmonic shuffle relationship,by studying the correlation properties of alternat-ing multiple zeta functions,the weighted mean identities of related forms such as(?)and(?)are obtained,Where k is any positive integer2.Using the generalized binomial coefficients and the related properties of harmonic numbers,we study the property of the form(?)and get the closed expression of(?)Where Hn(r)is the generalized harmonic number of order r.3.we study the congruence problem involving the sum of harmonic numbers and binomial coefficients,and get a new congruence formula for the product sum of binomial coefficients and harmonic numbers in modulo p3 and p2 for p>3.