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.

AOn page 212 of his lost notebook, Ramanujan defined a new class invariant λ n and constructed a table of values for λ n . The paper constructs a new class of series for 1\π associated with λ n . The new method also yields a new proof of the Borweins' general series for 1\π belonging to Ramanujan's ' theory of q # '.

In his lost notebook, Ramanujan offers several results related to the crank, the existence of which was first conjectured by F. J. Dyson and later established by G.E. Andrews and F.G. Garvan. Using an obscure identity found on p. 59 of the lost notebook, we provide uniform proofs of several congruences in the ring of formal power series for the generating function F (q) of cranks. All are found, sometimes in abbreviated form, in the lost notebook, and imply dissections of F (q). Consequences of our work are interesting new q-series identities and congruences in the spirit of Atkin and Swinnerton-Dyer.

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Abstract. In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R. Russell in 1887. We give a proof of Russell's main theorem and indicate the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields. Motivated by Russell's theorem, we state and prove its cubic analogue which allows us to construct Russell-type modular equations in the theory of signature 3.

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.

Abstract. A survey of Ramanujan's work on cranks in his lost notebook is given. We give evidence that Ramanujan was concentrating on cranks when he died, that is to say, the final problem on which Ramanujan worked was cranks -not mock theta functions.

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