{"id":1058,"date":"2024-08-19T06:43:00","date_gmt":"2024-08-19T06:43:00","guid":{"rendered":"https:\/\/worldculturepictorial.com\/wcp-blog\/?p=1058"},"modified":"2024-08-10T18:46:22","modified_gmt":"2024-08-10T18:46:22","slug":"mathematicians-make-breakthrough-in-distribution-of-prime-numbers-162-years-after-riemann-hypothesis","status":"publish","type":"post","link":"https:\/\/worldculturepictorial.com\/wcp-blog\/mathematicians-make-breakthrough-in-distribution-of-prime-numbers-162-years-after-riemann-hypothesis\/","title":{"rendered":"Mathematicians make breakthrough in distribution of prime numbers 162 years after Riemann hypothesis, 3 centuries after Gauss’s estimate"},"content":{"rendered":"
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\n\nMathematicians Clear Hurdle in Quest to Decode Primes<\/a>
\nPaul Nelson has solved the subconvexity problem, bringing mathematicians one step closer to understanding the Riemann hypothesis and the distribution of prime numbers.

\n\nIt\u2019s been 162 years since Bernhard Riemann posed a seminal question about the distribution of prime numbers. Despite their best efforts, mathematicians have made very little progress on the Riemann hypothesis. But they have managed to make headway on simpler related problems.

\n\nIn a paper posted in September, Paul Nelson of the Institute for Advanced Study has solved a version of the subconvexity problem, a kind of lighter-weight version of Riemann\u2019s question. The proof is a significant achievement on its own and teases the possibility that even greater discoveries related to prime numbers may be in store.

\n\nThe Riemann hypothesis and the subconvexity problem are important because prime numbers are the most fundamental \u2014 and most fundamentally mysterious \u2014 objects in mathematics. When you plot them on the number line, there appears to be no pattern to how they\u2019re distributed. But in 1859 Riemann devised an object called the Riemann zeta function \u2014 a kind of infinite sum \u2014 which fueled a revolutionary approach that, if proved to work, would unlock the primes\u2019 hidden structure. \u201cIt proves a result that a few years ago would have been regarded as science fiction,\u201d said Valentin Blomer of the University of Bonn.

\n\nHidden Structure of Prime Numbers Coming Into Focus<\/a>
\nJust as molecules are composed of atoms, in math, every natural number can be broken down into its prime factors\u2014those that are divisible only by themselves and 1. Mathematicians want to understand how primes are distributed along the number line, in the hope of revealing an organizing principle for the atoms of arithmetic.

\n\nFor 165 years, mathematicians seeking that structure have focused on the Riemann hypothesis. Proving it would offer a Rosetta Stone for decoding the primes\u2014as well as a $1 million award from the Clay Mathematics Institute. Now, in a preprint posted online on 31 May, Maynard and Larry Guth of the Massachusetts Institute of Technology have taken a step in this direction by ruling out certain exceptions to the Riemann hypothesis. The result is unlikely to win the cash prize, but it represents the first progress in decades on a major knot in math\u2019s biggest unsolved problem, and it promises to spark new advances throughout number theory.

\n\nPredicting exactly where the next prime will show up on the number line is challenging, but describing the cumulative abundance of primes over large intervals is surprisingly straightforward. In the late 1700s, at the age of 16, German mathematician Carl Friedrich Gauss saw that the frequency of prime numbers seems to diminish as they get bigger and posited that they scale according to a simple formula: the number of primes less than or equal to X is roughly X divided by the natural logarithm of X. Gauss\u2019s estimate has stood up impressively well. To the best mathematicians can tell, the actual number of primes bounces slightly above and below this curve up to infinity. That known primes follow such a simple formula so closely suggests the primes aren\u2019t completely random; there must be some deep connections governing where they appear.

\n\nBut mathematicians want to know exactly how well Gauss\u2019s guess holds up\u2014and why. In 1859, Bernhard Riemann, another renowned German mathematician, sought help from a different function, now called the Riemann zeta function. For inputs, the function takes complex numbers, which are a combination of real numbers and what mathematicians call \u201cimaginary\u201d ones: a normal number multiplied by the square root of \u20131. The function seems to capture the discrepancies between Gauss\u2019s curve and the real distribution of primes. The places where Riemann\u2019s function equals zero\u2014referred to as zeta zeros\u2014directly describe the fluctuating errors around Gauss\u2019s curve.

\n\nHere, Riemann made his famous conjecture: ignoring certain trivial solutions for negative inputs, all the zeta zeros should exist for inputs where the real part is one-half. If his hypothesis is true, it means the seemingly random fluctuations in the abundance of primes are bounded, leaving no big clumps or gaps in their distribution along the number line. Any proof of the Riemann hypothesis would be a window into the secret clockwork governing the primes\u2019 irregular pattern. It would offer a chance to \u201creverse-engineer the random number generator of the primes,\u201d says Maksym Radziwill, a mathematician at Northwestern University.

\n\n(unquote)

\n\nImage courtesy Claudio Rocchini\/Wikimedia Commons Cc-By<\/a>\n<\/p>\n","protected":false},"excerpt":{"rendered":"

(quote) Mathematicians Clear Hurdle in Quest to Decode Primes Paul Nelson has solved the subconvexity problem, bringing mathematicians one step closer to understanding the Riemann hypothesis and the distribution of prime numbers. It\u2019s been 162 years since Bernhard Riemann posed a seminal question about the distribution of prime numbers. DespiteRead More →<\/a><\/span><\/p>\n","protected":false},"author":17,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ep_exclude_from_search":false,"_vp_format_video_url":"","_vp_image_focal_point":[],"footnotes":""},"categories":[5,8,11,12],"tags":[26,27,129,39,35,31],"class_list":["entry","author-wcp-scientific-mind","post-1058","post","type-post","status-publish","format-standard","category-figures-and-facts","category-life-nature-society","category-science-and-technology","category-us-and-world","tag-facts","tag-figures","tag-math","tag-nature","tag-news","tag-science"],"_links":{"self":[{"href":"https:\/\/worldculturepictorial.com\/wcp-blog\/wp-json\/wp\/v2\/posts\/1058","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/worldculturepictorial.com\/wcp-blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/worldculturepictorial.com\/wcp-blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/worldculturepictorial.com\/wcp-blog\/wp-json\/wp\/v2\/users\/17"}],"replies":[{"embeddable":true,"href":"https:\/\/worldculturepictorial.com\/wcp-blog\/wp-json\/wp\/v2\/comments?post=1058"}],"version-history":[{"count":0,"href":"https:\/\/worldculturepictorial.com\/wcp-blog\/wp-json\/wp\/v2\/posts\/1058\/revisions"}],"wp:attachment":[{"href":"https:\/\/worldculturepictorial.com\/wcp-blog\/wp-json\/wp\/v2\/media?parent=1058"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/worldculturepictorial.com\/wcp-blog\/wp-json\/wp\/v2\/categories?post=1058"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/worldculturepictorial.com\/wcp-blog\/wp-json\/wp\/v2\/tags?post=1058"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}