In this paper we present some new binomial identities for multiple harmonic sums whose indices are the sequences ({1} a , c, {1} b ), ({2} a , c, {2} b ) and prove a number of congruences for these sums modulo a prime p. The congruences obtained allow us to find nice p-analogues of Leshchiner's series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo p. As a further application we provide a new proof of Zagier's formula for ζ * ({2} a , 3, {2} b ) based on a finite identity for partial sums of the zeta-star series.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3132 KH. HESSAMI PILEHROOD, T. HESSAMI PILEHROOD, AND R. TAURASOGlaisher [8] in 1900, and Lehmer [21] in 1938, proved that even the multiple harmonic sums H p−1 (m) modulo a higher power of a prime p ≥ m + 3 are related to the Bernoulli numbers:The systematic study of MZVs began in the early 1990s with the works of Hoffman [16] and Zagier [33]. The set of the MZVs has a rich algebraic structure given by the shuffle and the stuffle (harmonic shuffle or quasi-shuffle) relations. These follow from the representation of multiple zeta values in terms of iterated integrals and harmonic sums, respectively. There are many conjectures concerning multiple zeta values, and despite some recent progress, lots of open questions still remain to be answered. Licensed to New York Univ, Courant Inst. Prepared on Tue Feb 3 02:49:34 EST 2015 for download from IP 128.122.253.228. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use NEW PROPERTIES OF MULTIPLE HARMONIC SUMS 3133Let Z w denote the Q-vector space spanned by the set of multiple zeta values ζ(s 1 , . . . , s r ) with s r ≥ 2 and the total weight w = s 1 + · · · + s r , and let Z denote the Q-vector space spanned by all multiple zeta values over Q. A conjecture of Zagier [33] states that the dimension of the Q-vector space Z w is given by the Perrin numbers d w defined for w ≥ 3 by the recurrencewith the initial conditions d 0 = 1, d 1 = 0, d 2 = 1. The upper bound dim Z w ≤ d w was proved independently by Goncharov [9] and Terasoma [28].It is easy to see that the Perrin number d w is equal to the number of multiple zeta values ζ(s 1 , . . . , s r ) with s 1 + · · · + s r = w and each s j ∈ {2, 3}. While investigating the deep algebraic structure of Z, Hoffman [17] conjectured that the MZVs ζ(s 1 , . . . , s r ) of weight w with s j ∈ {2, 3} span the Q-space Z w . Very recently, this conjecture was proved using motivic ideas by Brown [4]. So the main problem which remains open is proving that the numbers ζ(s 1 , . . . , s r ) with s j ∈ {2, 3} are linearly independent over Q.According to Zagier's conjecture, a basis for Z w for 2 ≤

We generalize Sondow's (ir)rationality criteria for Euler's constant and give necessary and sufficient conditions for irrationality of generalized Euler's constant g a ; as well as obtain new asymptotic formulas for computing g a : The proof is based on constructing linear forms in 1; g a and logarithms of rational numbers. r 2004 Elsevier Inc. All rights reserved.

We prove approximation formulas for the logarithms of some infinite products, in particular, for Euler's constant γ , log 4 π and log σ , where σ is Somos's quadratic recurrence constant, in terms of classical Legendre polynomials and partial sums of their series expansions. We also give conditional irrationality and linear independence criteria for these numbers. The main tools are Euler-type integrals, hypergeometric series, and Laplace method.

Abstract. In recent years, there has been intensive research on the Q-linear relations between multiple zeta (star) values. In this paper, we prove many families of identities involving the q-analog of these values, from which we can always recover the corresponding classical identities by taking q → 1. The main result of the paper is the duality relations between multiple zeta star values and Euler sums and their q-analogs, which are generalizations of the Two-one formula and some multiple harmonic sum identities and their q-analogs proved by the authors recently. Such duality relations lead to a proof of the conjecture by Ihara et al. that the Hoffman ⋆-elements ζ ⋆ (s1, . . . , sr) with si ∈ {2, 3} span the vector space generated by multiple zeta values over Q.