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

It is known by a formula of Hasse-Sondow that the Riemann zeta function isgiven, for any $ s=\sigma+it \in \mathbb{C}$, by $ \sum_{n=0}^{\infty}\widetilde{A}(n,s)$ where $$ \widetilde{A}(n,s):=\frac{1}{2^{n+1}(1-2^{1-s})}\sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{(k+1)^s} .$$ We prove the followingapproximate functional equation for the Hasse-Sondow presentation: For $ \vertt \vert = \pi xy $ and $ 2y \neq (2N-1)\pi $ then $$ \zeta(s)= \sum_{n \leq x }\widetilde{A}(n,s)+\frac{\chi(s)}{1-2^{s-1}} \left (\sum_{k \leq y}(2k-1)^{s-1} \right ) +O \left (e^{-\omega(x,y) t} \right ), $$ where $ 0<\omega(x,y)$ is a certain transcendental number determined by $ x$ and $ y$. Acentral feature of our new approximate functional equation is that its errorterm is of exponential rate of decay. The proof is based on a study, via saddlepoint techniques, of the asymptotic properties of the function $$\widetilde{A}(u,s):= \frac{1}{2^{u+1} (1-2^{1-s}) \Gamma(s)} \int_{0}^{\infty}\left ( e^{-w} \left ( 1- e^{-w} \right)^u \right ) w^{s-1} dw,$$ and integralsrelated to it.

An approximate functional equation for the Riemann zeta function with exponentially decaying error