English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The Minkowski question mark function is a rich object which can be exploredfrom the perspective of dynamical systems, complex dynamics, metric numbertheory, multifractal analysis, transfer operators, integral transforms, and asa function itself via analysis of continued fractions and convergents. Ourpermanent target, however, was to get arithmetic interpretation of the momentsof ?(x) (which are relatives of periods of Maass wave forms) and to relate thefunction ?(x) to certain modular objects. In this paper we establish this link,embedding ?(x) not into the modular-world itself, but into a space of functionswhich are generalizations and which we call mean-modular forms. For thispurpose we construct a wide class of measures, and also investigate modularforms for congruence subgroups which additionally satisfy the three termfunctional equation. From this perspective, the modular forms for the wholemodular group as well as the Stieltjes transform of ?(x) (the dyadic periodfunction) minus the Eisenstein series of weight 2 fall under the same uniformdefinition. The main result is the construction of the canonical isomorphismbetween the spaces of quasi-modular forms and mean-modular forms. This givesunexpected Minkowski question mark function-related interpretation ofquasi-modular forms.

The Minkowski $?(x)$ function, a class of singular measures, quasi-modular and mean-modular forms. I