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

We show that there are an infinite number of Riemann zeros on the criticalline, enumerated by the positive integers $n=1,2,\dotsc$, whose ordinates canbe obtained as the solution of a new transcendental equation that depends onlyon $n$. Under weak assumptions, we show that the number of such zeros alreadysaturates the counting formula for the numbers of zeros on the entire criticalstrip. These results thus constitute a concrete proposal toward verifying theRiemann hypothesis. We perform numerical analyses of the exact equation, andits asymptotic limit of large ordinate. The starting point is an explicitanalytical formula for an approximate solution to the exact equation in termsof the Lambert $W$ function. In this way, we neither have to use Gram points ordeal with violations of Gram's law. Our numerical approach thus constitutes anovel method to compute the zeros. Employing these numerical solutions, weverify that solutions of the asymptotic version are accurate enough to confirmMontgomery's and Odlyzko's pair correlation conjectures and also to reconstructthe prime number counting function.

Statistical and other properties of Riemann zeros based on an explicit equation for the $n$-th zero on the critical line