300 BCE
Euclid's Proof of Infinitude of Primes
Euclid provides the first known proof that there are infinitely many prime numbers, a foundational result in number theory. #math #history
Chronological Storyline (40 Milestones)
1735 CE
Euler Solves the Basel Problem
Euler shows that the sum of reciprocals of squares equals π²/6, establishing the value of ζ(2) and launching analytic number theory. #math #numbertheory
By Cmglee - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=168287114
1737 CE
Euler's Product Formula for the Zeta Function
Euler introduces the product over primes for the zeta function, linking prime numbers to complex analysis and foreshadowing the Riemann zeta function. #math #zeta
1742 CE
Goldbach's Conjecture Proposed
Goldbach conjectures that every even integer greater than 2 can be expressed as the sum of two primes, an unsolved problem that drives analytic number theory. #math #conjecture
By Christian Goldbach - http://www.mscs.dal.ca/~joerg/pic/g-letter.jpg, Public domain, https://commons.wikimedia.org/w/index.php?curid=1721422
1798 CE
Legendre's Conjecture
Legendre conjectures that there is always a prime between n² and (n+1)², a statement still unproven today. #math #primes
1800 CE
Gauss's Heuristic on Prime Distribution
Gauss conjectures that the prime counting function π(x) is asymptotically x / log x, laying the groundwork for the Prime Number Theorem. #math #primes
1837 CE
Dirichlet's Theorem on Arithmetic Progressions
Dirichlet proves that there are infinitely many primes in any arithmetic progression a mod q with gcd(a,q)=1, introducing Dirichlet L-functions. #math #analytic
By Unknown author - http://www.york.ac.uk/depts/maths/histstat/people/, Public domain, https://commons.wikimedia.org/w/index.php?curid=2914309
1852 CE
Chebyshev's Bounds on Prime Distribution
Chebyshev provides the first rigorous bounds for π(x), showing 0.921x/log x < π(x) < 1.105x/log x for large x. #math #numbertheory
By Atelier Nadar. Photographe - Bibliothèque nationale de France, Public domain, https://commons.wikimedia.org/w/index.php?curid=195251267
1859 CE
Riemann's Memoir on the Zeta Function
Riemann publishes 'On the Number of Primes Less Than a Given Magnitude', introducing the Riemann zeta function in the complex plane and formulating the Riemann Hypothesis. #math #zeta #riemannhypothesis
By Georg Friedrich Bernhard Riemann, 1826-1866. - Monatsberichte der Berliner Akademie, November 1859, Public domain, https://commons.wikimedia.org/w/index.php?curid=21321366
1874 CE
Mertens' Theorems
Mertens proves three theorems on sums and products over primes, including the asymptotic product of (1-1/p) ~ e^{-γ}/log x. #math #primes
1896 CE
Proof of the Prime Number Theorem
Hadamard and de la Vallée Poussin independently prove that π(x) ~ x / log x using complex analysis of the zeta function. #math #primes
1904 CE
Voronoi's Summation Formula
Voronoi derives a general summation formula for divisor sums, a powerful tool in analytic number theory. #math #analytic
1913 CE
Ramanujan's Master Theorem
Ramanujan states a general theorem for evaluating integrals and series, widely used in analytic number theory. #math #ramanujan
By Srinavasa Ramanujan - http://www.imsc.res.in/~rao/ramanujan/, Public domain, https://commons.wikimedia.org/w/index.php?curid=67637326
1914 CE
Littlewood's Theorem on π(x) - li(x)
Littlewood proves that the difference π(x) - li(x) changes sign infinitely often, a surprising result on prime distribution. #math #primes
1916 CE
Ramanujan's Contributions to the Tau Function
Ramanujan introduces the tau function and its connections to the zeta function, inspiring later deep work on modular forms. #math #ramanujan
By Unknown author - https://mss-cat.trin.cam.ac.uk/manuscripts/uv/view.php?n=Add.Ms.a.94.7#?c=0&m=0&s=0&cv=6&xywh=-2849%2C-273%2C9238%2C5258, CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=153767257
1918 CE
Hardy-Ramanujan Asymptotic for Partitions
Hardy and Ramanujan derive an asymptotic formula for the partition function p(n), founding the circle method. #math #partitions )#Hardy%E2%80%93Ramanujan_asymptotic_formula
By R. A. Nonenmacher - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=4766072
1926 CE
Kloosterman Sums Introduced
Kloosterman introduces exponential sums now named after him, crucial for many problems in analytic number theory. #math #exponentials
1927 CE
van der Corput's Method for Exponential Sums
van der Corput develops a systematic method to bound exponential sums, fundamental to analytic number theory. #math #exponentialsums
1932 CE
Riemann–Siegel Formula Discovered
Siegel publishes Riemann's unpublished formula for computing ζ(1/2+it), enabling efficient numerical computation. #math #zeta
1937 CE
Ingham's Theorem on Prime Gaps
Ingham shows that for large n, there exists a prime between n³ and (n+1)³, using the zeta function. #math #primes
1937 CE
Vinogradov's Three-Prime Theorem
Vinogradov proves that every sufficiently large odd number is the sum of three primes, a landmark in additive number theory. #math #goldbach
By Unknown author - Original publication: Газета «Алтайская правда» №120 (7277) от 19 июня 1945 годаImmediate source: warheroes.ru, Public domain, https://commons.wikimedia.org/w/index.php?curid=95028677
1939 CE
Cartwright's Work on Dirichlet Series
Mary Cartwright studies zeros of Dirichlet series related to the zeta function, advancing the theory of entire functions. #math #women
By by Elliott & Fry, bromide print, 1950, NPG x86637 - https://www.npg.org.uk/collections/search/person/mp70948/dame-mary-lucy-cartwright, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=45122303
1944 CE
Linnik's Theorem on the Least Prime
Linnik proves that there exists a prime in every arithmetic progression with modulus q, bounded by a power of q. #math #primes
1949 CE
Selberg and Erdős Elementary Proof of PNT
Selberg and Erdős independently give an elementary proof of the Prime Number Theorem, without complex analysis. #math #primes
1952 CE
Weil's Explicit Formula
Weil generalizes the Riemann–von Mangoldt formula, relating sums over primes to sums over zeta zeros. #math #zeta )
1956 CE
Selberg's Trace Formula
Selberg generalizes the Poisson summation formula to non-Euclidean spaces, connecting geometry, analysis, and number theory. #math #traceformula
1965 CE
Bombieri–Vinogradov Theorem
Bombieri proves a mean value theorem for primes in arithmetic progressions, a key tool in sieve theory. #math #primes
1966 CE
Chen's Theorem on Goldbach's Conjecture
Chen Jingrun proves that every sufficiently large even number is the sum of a prime and a number with at most two prime factors. #math #china
By user:Iamdavidtheking - File:A statue of Chen Jingrun.JPG, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=8001620
1973 CE
Montgomery's Pair Correlation Conjecture
Montgomery conjectures that the pair correlation of zeros of the Riemann zeta function matches that of random Hermitian matrices. #math #zeta #randommatrix
By Renate Schmid - Mathematisches Institut Oberwolfach (MFO), https://opc.mfo.de/detail?photoID=10358, CC BY-SA 2.0 de, https://commons.wikimedia.org/w/index.php?curid=4331529
1974 CE
Deligne Proves the Weil Conjectures
Deligne proves the Riemann hypothesis for curves over finite fields, using étale cohomology, with profound influence on analytic number theory. #math #algebraic
1976 CE
Goldfeld's Theorem on Class Numbers
Goldfeld uses analytic number theory to prove that the class number of imaginary quadratic fields grows slowly, resolving a Gauss problem. #math #classnumber
1989 CE
Conrey's Theorem on Zeta Zeros
Conrey proves that at least 2/5 of the non-trivial zeros of the Riemann zeta function lie on the critical line. #math #riemannhypothesis
1990 CE
Odlyzko's Numerical Computations of Zeta Zeros
Odlyzko computes millions of zeta zeros, providing strong numerical evidence for Montgomery's pair correlation conjecture. #math #computing
By Konrad Jacobs, Erlangen, Copyright is MFO - Mathematisches Forschungsinstitut Oberwolfach,https://opc.mfo.de/detail?photo_id=3142, CC BY-SA 2.0 de, https://commons.wikimedia.org/w/index.php?curid=12349092
1998 CE
Sarnak and Random Matrix Theory
Sarnak connects the statistical distribution of zeta zeros to random matrix theory, advancing the understanding of the Riemann Hypothesis. #math #randommatrix
By VOA - http://www.voachinese.com/media/video/i-america-math-zhang-yitang-20131204/1803128.html, Public domain, https://commons.wikimedia.org/w/index.php?curid=111597486
2004 CE
Green–Tao Theorem on Primes in Arithmetic Progressions
Ben Green and Terence Tao prove that the primes contain arbitrarily long arithmetic progressions. #math #primes
2013 CE
Zhang's Bounded Gaps Between Primes
Yitang Zhang proves that there are infinitely many pairs of primes separated by less than 70 million. #math #primes
By VOA - http://www.voachinese.com/media/video/i-america-math-zhang-yitang-20131204/1803128.html, Public domain, https://commons.wikimedia.org/w/index.php?curid=33336006
2013 CE
Helfgott's Proof of the Ternary Goldbach Conjecture
Harald Helfgott proves that every odd integer greater than 5 is the sum of three primes. #math #goldbach
2013 CE
Maynard's Improvement on Prime Gaps
James Maynard improves Zhang's bound, showing that there are infinitely many pairs of primes with gap at most 600, and extends the result to k-tuples. #math #primes )
By Petra Lein, Copyright is MFO - Mathematisches Forschungsinstitut Oberwolfach, https://owpdb.mfo.de/detail?photo_id=18228, CC BY-SA 2.0 de, https://commons.wikimedia.org/w/index.php?curid=43553945
2015 CE
Tao Solves Erdős Discrepancy Problem
Terence Tao uses analytic number theory to prove the Erdős discrepancy problem, showing that any sequence with bounded partial sums must have bounded discrepancy. #math #discrepancy
2020 CE
Hua Luogeng's Legacy in Analytic Number Theory
Hua Luogeng's work on Waring's problem and his contributions to the theory of exponential sums continue to influence modern analytic number theory. #math #china
By 牛畏予 - http://baike.baidu.com/albums/6351/6351.html#0$e78c65898b37b4d30e244463, Public domain, https://commons.wikimedia.org/w/index.php?curid=17315495