For positive integers α 1 , α 2 , . . . , α r with α r 2, the multiple zeta value or r-fold Euler sum is defined asThere is a celebrated sum formula among multiple zeta values as |α|=m ζ(α 1 , α 2 , . . . , α r + 1) = ζ(m + 1), where α 1 , α 2 , . . . , α r range over all positive integers with |α| = α 1 + α 2 + · · · + α r = m in the summation. In this paper, we shall prove that for all positive integers m and q with m q, and a nonnegative integer p,When p = 0 and q = r, this is precisely the sum formula. Such a formula can be used to determine explicitly, some multiple zeta values of lower weights more precisely than the sum formula.

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For positive integers α 1 , α 2 , . . . , αr with αr 2, the multiple zeta value or r-fold Euler sum is defined bywhere |α| = α 1 + α 2 + • • • + αr and r are the weight and depth of ζ(α 1 , α 2 , . . . , αr) respectively. By the general theorem given in [11], the multiple zeta value ζ(α 1 , α 2 , . . . , αr) can be expressed as a rational linear combination of products of multiple zeta values of lower depth if its depth and weight are of different parity. In other words, when the sum of its depth and weight is odd. However, there are still some exceptions for quadruple Euler sums. As conjectured in [6], a quadruple Euler sum with even weight exceeding 14 can be expressed as a rational linear combination of products of multiple zeta values of depth 1, 2 and 3 if and only if it is one of the following forms: ζ(1, a, b, a), ζ(b, 1, a, a), ζ(b, b, 1, a) or ζ(a, b, b, a) with a = b or b = 1. In this paper, we shall evaluate these quadruple Euler sums of even weight by the identities among multiple zeta values with variables and relation obtained from the stuffle formula of two multiple zeta values.

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Bernoulli numbers appear as special values of zeta functions at integers and identities relating the Bernoulli numbers follow a s a consequence of properties of the corresponding zeta functions. The most famous example is that of the special values of the Riemann zeta function and the Bernoulli identities due to Euler. In this paper we introduce a general principle for producing Bernoulli identities and apply it to zeta functions considered by Shintani, Zagier and Eie. Our results include some of the classical results of Euler and Ramanujan. Kummer's congruences play important roles in the investigation of padic interpolation of the classical Riemann zeta function. It asserts congruence relations among Bernoulli numbers,i.e.

Printed in Singapore PrefaceThis is a record of my research on multiple zeta values since 2007. In a series of lecture notes used in classes or in seminars on number theory, I provided detailed proofs for both newly found and well-known results. These lecture notes were extremely useful and helpful to my graduate students in learning number theory and writing their thesis. Some particularly interesting sections were published as research papers. These lecture notes are now reworked into a book, which I hope will also be helpful for researchers. This book contains the following important topics concerning multiple zeta values and applications.1. The duality theorem, the sum formula and the restricted sum formula of multiple zeta values.2. Shuffle relations produced from shuffle products of multiple zeta values.3. Double weighted sum formulas of multiple zeta values.4. Applications of shuffle products in combinatorics.5. Combinatorial identities of convolution type.6. Generalizations of Pascal identity.

In this paper, we compute shuffle relations from multiple zeta values of the form ζ({1}m-1, n+1) or sums of multiple zeta values of fixed weight and depth. Some interesting weighted sum formulas are obtained, such as [Formula: see text] where m and k are positive integers with m ≥ 2k. For k = 1, this gives Ohno–Zudilin's weighted sum formula.