Transcendental values of certain Eichler integrals

Abstract

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.

Concerning dense subideals in commutative Banach algebras

“…We were disappointed that we were unable to to deduce a similar theorem concerning P k (z), however we are hopeful that the approach outlined in Section 3 might eventually succeed in this case. An additional avenue might involve studying the zeros of a truncated version of the right-hand side of (7). Notice that the zeros of P k (z) are very well approximated by the zeros of Thus, it should be a worthwhile endeavor to study the zeros of these auxiliary functions.…”

“…Not surprinsigly, this formula is also mentioned in works dealing with the irrationality of ζ(3) [11]. Notice that Gun, Murty and Rath used (7) to express odd zeta values in terms of Eichler integrals [7]. Now consider the case when k = 2.…”

“…Acknowledgements The authors thank Wadim Zudilin for the interesting discussions, encouragement and helpful comments, and M. Ram Murty for calling our attention to [7]. A portion of this research was carried out while Mat Rogers was visiting the University of Georgia.…”

A Number Field Extension of a Question of Milnor

“…This is the content of Theorem 2.2 in the next section. See also [4] and [6] for a modular interpretation of the conjectural transcendence of the normalised values ζ(2d + 1)/π 2d+1 . It will be evident that considering the extended Milnor conjecture to a number field K comes with a caveat, namely it depends on the arithmetic of K (for instance compare Theorem 3.3 with Corollary 4.2).…”

Feynman integrals and critical modular $L$-values

“…The transformation (16) is obvious from the definition of F 3 (τ ) and (17) is a special case of Grosswald's formula [18,19]. Using the integral expression (15) for each term in (18) and performing a change of variable in each integral, we obtain…”