[{"id":248019,"date":"2026-10-07T17:29:28","date_gmt":"2026-10-07T22:29:28","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=248019"},"modified":"2026-10-07T17:29:28","modified_gmt":"2026-10-07T22:29:28","slug":"consequences-of-qrh","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/07\/consequences-of-qrh\/","title":{"rendered":"Consequences of progress toward the Riemann Hypothesis"},"content":{"rendered":"

The Riemann Hypothesis (RH) is the conjecture that all the zeros of the Riemann zeta function \u03b6(s<\/em>) in the critical strip, i.e. the region of the complex plane with real part between 0 and 1, have real part equal to \u00bd.<\/p>\n

The Quasi Riemann Hypothesis (QRH) says that there exists a constant \u03b8 < 1 such that no zeros of \u03b6(s<\/em>) have real part greater than \u03b8. OpenAI<\/a> has published a paper claiming QRH with \u03b8 = 7\/8.<\/p>\n

The RH is so important to number theory that even partial results can have big consequences. This post will focus on one consequence: the error term in the Prime Number Theorem.<\/p>\n

The Prime Number Theorem says that \u03c0(x<\/em>), the number of primes less than x<\/em>, is asymptotically equal to Li(x<\/em>). We’d like to know more specifically at what rate \u03c0(x<\/em>) approaches Li(x<\/em>).<\/p>\n

The best known result before the QRH announcement was<\/p>\n

\"\\pi(x)<\/p>\n

If the QRH holds for some \u03b8, such as OpenAI’s assertion that \u03b8 = 7\/8,<\/p>\n

\"\\pi(x)
\nIf RH holds, \u03b8 = \u00bd.<\/p>\n

Incidentally, you may have seen the Prime Number Theorem stated with\u00a0x<\/em>\/log(x<\/em>) rather than Li(x<\/em>). These two functions are asymptotically equal, so they give the same theorem, if you’re not interested in quantifying the rate of convergence. The function Li(x<\/em>) gives better error bounds.<\/p>\n","protected":false},"excerpt":{"rendered":"

The Riemann Hypothesis (RH) is the conjecture that all the zeros of the Riemann zeta function \u03b6(s) in the critical strip, i.e. the region of the complex plane with real part between 0 and 1, have real part equal to \u00bd. The Quasi Riemann Hypothesis (QRH) says that there exists a constant \u03b8 < 1 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[94],"class_list":["post-248019","post","type-post","status-publish","format-standard","hentry","category-math","tag-number-theory"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"Consequences of progress toward the Riemann Hypothesis","description":"Consequences of the recent announcement of a proof of the Quasi Riemann Hypothesis","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/07\/consequences-of-qrh\/","robots":"max-image-preview:large","keywords":"number theory","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Consequences of progress toward the Riemann Hypothesis","og:description":"Consequences of the recent announcement of a proof of the Quasi Riemann Hypothesis","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/07\/consequences-of-qrh\/","article:published_time":"2026-10-07T22:29:28+00:00","article:modified_time":"2026-10-07T22:29:28+00:00","twitter:card":"summary","twitter:title":"Consequences of progress toward the Riemann Hypothesis","twitter:description":"Consequences of the recent announcement of a proof of the Quasi Riemann Hypothesis","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"248019","title":null,"description":"Consequences of the recent announcement of a proof of the Quasi Riemann Hypothesis","keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"created":"2026-10-07 21:54:44","updated":"2026-10-07 22:29:28","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":"

\n\t\t\tHome<\/a>\n\t\t<\/span>\u00bb<\/span>\n\t\t\tMath<\/a>\n\t\t<\/span>\u00bb<\/span>\n\t\t\tConsequences of progress toward the Riemann Hypothesis\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/www.johndcook.com\/blog\/"},{"label":"Math","link":"https:\/\/www.johndcook.com\/blog\/category\/math\/"},{"label":"Consequences of progress toward the Riemann Hypothesis","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/07\/consequences-of-qrh\/"}],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248019","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/comments?post=248019"}],"version-history":[{"count":1,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248019\/revisions"}],"predecessor-version":[{"id":248020,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248019\/revisions\/248020"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=248019"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=248019"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=248019"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}},{"id":248015,"date":"2026-10-07T16:41:47","date_gmt":"2026-10-07T21:41:47","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=248015"},"modified":"2026-10-07T16:45:37","modified_gmt":"2026-10-07T21:45:37","slug":"faster-fourier-transform","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/07\/faster-fourier-transform\/","title":{"rendered":"Faster Fourier Transform"},"content":{"rendered":"

The Fast Fourier Transform (FFT) algorithm can compute the discrete Fourier transform of a sequence of length\u00a0n<\/em> in time<\/p>\n

O<\/em>(n<\/em> log\u00a0n<\/em>).<\/p>\n

OpenAI<\/a> recently posted a paper saying there is an algorithm that could compute the discrete Fourier transform in<\/p>\n

O<\/em>(n<\/em> (log\u00a0n<\/em>)1 \u2212 \u03b5<\/sup>)<\/p>\n

time for \u03b5 = 10\u221213<\/sup>.<\/p>\n

This result is amazing. It seemed that O<\/em>(n<\/em> log\u00a0n<\/em>) was as good as you could do, which it provably is for sorting algorithms.<\/p>\n

The result is also of absolutely no practical value, for now. But since the theorem shows that our assumptions were wrong regarding what we thought was possible, however slightly, maybe we’re in for further surprises. Maybe the \u03b5 crack will grow. It wouldn’t be the first time.<\/p>\n

Related posts<\/h2>\n