We study the transcendence of certain Eichler integrals associated to Eisenstein series and more generally to modular forms using functional identities due to Ramanujan, Grosswald, Weil et al.The special values of such integrals at algebraic points in the upper half-plane are linked to Riemann zeta values at odd positive integers.
Abstract. In this paper, we investigate a conjecture due to S. and P. Chowla and its generalization by Milnor. These are related to the delicate question of non-vanishing of L-functions associated to periodic functions at integers greater than 1. We report on some progress in relation to these conjectures. In a different vein, we link them to a conjecture of Zagier on multiple zeta values and also to linear independence of polylogarithms.
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Let ψ(x) denote the digamma function. We study the linear independence of ψ(x) at rational arguments over algebraic number fields. We also formulate a variant of a conjecture of Rohrlich concerning linear independence of the log gamma function at rational arguments and report on some progress. We relate these conjectures to non-vanishing of certain L-series.
For an abelian group G, the Davenport constant D(G) is defined to be\ud
the smallest natural number k such that any sequence of k elements in G has a nonempty\ud
subsequence whose sum is zero (the identity element). Motivated by some recent\ud
developments around the notion of Davenport constant with weights, we study them in\ud
some basic cases.We also define a new combinatorial invariant related to (Z/nZ)^d , more\ud
in the spirit of some constants considered by Harborth and others and obtain its exact\ud
value in the case of (Z/nZ)^2 where n is an odd integer
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