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We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing modular invariant functions based on iterated integrals of modular forms. The construction will be based on an extension of higher-order modular forms which, in contrast to the standard higher-order forms, applies to general Fuchsian groups of the first kind and, as such, is of independent interest. [ABSTRACT FROM AUTHOR]

We consider iterated integrals of logζ(s) on certain vertical and horizontal lines. Here, the function ζ(s) is the Riemann zeta-function. It is a well-known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of ∫0tlogζ(1/2 + it′)⁢ 𝑑t′ under the Riemann Hypothesis. Moreover, we show that, for any m ≥ 2, the denseness of the values of an m-times iterated integral on the critical line is equivalent to the Riemann Hypothesis. [ABSTRACT FROM AUTHOR]

We study weight one specializations of the Euler system of Beilinson–Flach elements introduced by Kings, Loeffler and Zerbes, with a view towards a conjecture of Darmon, Lauder and Rotger relating logarithms of units in suitable number fields to special values of the Hida–Rankin p-adic L-function. We show that the latter conjecture follows from expected properties of Beilinson–Flach elements and prove the analogue of the main theorem of Castella and Hsieh about generalized Kato classes. [ABSTRACT FROM AUTHOR]