--- title: What does the Riemann zeta function have to do with the... description: A deep mathematical exploration of how the Riemann zeta function connects to the distribution of prime numbers. Starting from Euler's proof that the sum of... canonical: https://daily.dev/posts/what-does-the-riemann-zeta-function-have-to-do-with-the-distribution-of-the-primes--dx3wpxe9x twitter:card: summary_large_image twitter:site: @dailydotdev og:type: website og:site_name: daily.dev og:title: What does the Riemann zeta function have to do with the distribution of the primes? | daily.dev og:description: A deep mathematical exploration of how the Riemann zeta function connects to the distribution of prime numbers. Starting from Euler's proof that the sum of... og:url: https://daily.dev/posts/what-does-the-riemann-zeta-function-have-to-do-with-the-distribution-of-the-primes--dx3wpxe9x og:image: https://api.daily.dev/og/posts/dX3wPXE9X.png og:image:alt: What does the Riemann zeta function have to do with the distribution of the primes? og:image:width: 1200 og:image:height: 630 og:locale: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # What does the Riemann zeta function have to do with the distribution of the primes? **[Hacker News](https://daily.dev/sources/hn)** · 16 min read · 0 upvotes · 0 comments ## Summary A deep mathematical exploration of how the Riemann zeta function connects to the distribution of prime numbers. Starting from Euler's proof that the sum of reciprocals of primes diverges, the post walks through the Euler product formula, logarithmic techniques, and Taylor series to establish that primes are not too sparse. It then derives Riemann's explicit formula π(x) ≈ Li(x) − Σ Li(x^ρ), showing how the zeroes of the zeta function determine the accuracy of Gauss's prime counting approximation. The post concludes by explaining why the Riemann Hypothesis — that all non-trivial zeroes have real part 1/2 — would make Gauss's approximation as accurate as possible. ## Full article daily.dev links to this article rather than hosting it. Read it at the original source: ## Similar posts on daily.dev - [Using calculus to do number theory](https://daily.dev/posts/using-calculus-to-do-number-theory-cdd3hm6ig) · Hacker News · 1 upvotes · 0 comments - [The Geometry of Uncertainty: A Fresh Perspective on Probability Distributions](https://daily.dev/posts/the-geometry-of-uncertainty-a-fresh-perspective-on-probability-distributions-wuippty97) · Medium · 0 upvotes · 0 comments --- Tags: [#math](https://daily.dev/tags/math) [View this post on daily.dev](https://daily.dev/posts/what-does-the-riemann-zeta-function-have-to-do-with-the-distribution-of-the-primes--dx3wpxe9x) ```json {"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://daily.dev/#organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180},"sameAs":["https://twitter.com/dailydotdev","https://github.com/dailydotdev","https://www.linkedin.com/company/daily-dev-ltd"]},{"@type":"WebSite","@id":"https://daily.dev/#website","url":"https://daily.dev","name":"daily.dev","publisher":{"@id":"https://daily.dev/#organization"},"potentialAction":{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://daily.dev/search?q={search_term_string}"},"query-input":"required name=search_term_string"}}]} {"@context":"https://schema.org","@type":"TechArticle","headline":"What does the Riemann zeta function have to do with the distribution of the primes?","url":"https://daily.dev/posts/what-does-the-riemann-zeta-function-have-to-do-with-the-distribution-of-the-primes--dx3wpxe9x","mainEntityOfPage":{"@type":"WebPage","@id":"https://daily.dev/posts/what-does-the-riemann-zeta-function-have-to-do-with-the-distribution-of-the-primes--dx3wpxe9x"},"datePublished":"2026-07-20T14:47:00.919Z","dateModified":"2026-07-20T14:47:23.671Z","description":"A deep mathematical exploration of how the Riemann zeta function connects to the distribution of prime numbers. Starting from Euler's proof that the sum of...","image":"https://media.daily.dev/image/upload/s--P4t4XyoV--/f_auto/v1722860399/public/Placeholder%2001","thumbnailUrl":"https://media.daily.dev/image/upload/s--P4t4XyoV--/f_auto/v1722860399/public/Placeholder%2001","isAccessibleForFree":true,"articleSection":"Hacker News","inLanguage":"en","publisher":{"@type":"Organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180}},"author":{"@type":"Organization","name":"Hacker News","logo":"https://media.daily.dev/image/upload/t_logo,f_auto/v1/logos/hn","url":"https://daily.dev/sources/hn"},"commentCount":0,"discussionUrl":"https://daily.dev/posts/what-does-the-riemann-zeta-function-have-to-do-with-the-distribution-of-the-primes--dx3wpxe9x","interactionStatistic":[{"@type":"InteractionCounter","interactionType":{"@type":"LikeAction"},"userInteractionCount":0},{"@type":"InteractionCounter","interactionType":{"@type":"CommentAction"},"userInteractionCount":0}],"keywords":"math","timeRequired":"PT16M"} {"@context":"https://schema.org","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://daily.dev"},{"@type":"ListItem","position":2,"name":"Hacker News","item":"https://daily.dev/sources/hn"},{"@type":"ListItem","position":3,"name":"What does the Riemann zeta function have to do with the distribution of the primes?"}]} ```