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 Plus annual

Presents the access of premium content as premium feature

Premium Content

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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For any elliptic curve $E$ over $k\subset \Bbb R$ with $E({\Bbb C})={\BbbC}^\times/q^{\Bbb Z}$, $q=e^{2\pi iz}, \Im(z)>0$, we study the $q$-average$D_{0,q}$, defined on $E({\Bbb C})$, of the function $D_0(z) = \Im(z/(1-z))$.Let $\Omega^+(E)$ denote the real period of $E$. We show that there is arational function $R \in {\Bbb Q}(X_1(N))$ such that for any non-cuspidal realpoint $s\in X_1(N)$ (which defines an elliptic curve $E(s)$ over $\Bbb R$together with a point $P(s)$ of order $N$), $\pi D_{0,q}(P(s))$ equals$\Omega^+(E(s))R(s)$. In particular, if $s$ is $\Bbb Q$-rational point of$X_1(N)$, a rare occurrence according to Mazur, $R(s)$ is a rational number.

Eisenstein series of weight one, $q$-averages of the $0$-logarithm and periods of elliptic curves