{ "id": "epcbaf6x6s", "slug": "sarvadaman-chowla", "title": "Sarvadaman Chowla", "updated": "2026-10-21", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.scientists", "label": "Physical and mathematical scientists", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists" }, { "id": "physical.scientists.mathematics-statistics", "label": "Mathematicians and statisticians", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.mathematics-statistics" }, { "id": "physical.scientists.mathematics-statistics.number-theorists", "label": "Number theorists", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.mathematics-statistics.number-theorists" }, { "id": "physical.scientists.mathematics-statistics.number-theorists.analytic-number-theorists", "label": "Analytic number theorists", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.mathematics-statistics.number-theorists.analytic-number-theorists" } ], "geo": [ { "id": "geo.us.t1946.physical.scientists.mathematics-statistics.number-theorists", "label": "United States · 1946 to 2000: Number theorists", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical.scientists.mathematics-statistics.number-theorists", "path": [ { "id": "geo.us", "label": "United States", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us" }, { "id": "geo.us.t1946", "label": "United States · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946" }, { "id": "geo.us.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical" }, { "id": "geo.us.t1946.physical.scientists", "label": "Physical and mathematical scientists", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical.scientists" }, { "id": "geo.us.t1946.physical.scientists.mathematics-statistics", "label": "Mathematicians and statisticians", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical.scientists.mathematics-statistics" }, { "id": "geo.us.t1946.physical.scientists.mathematics-statistics.number-theorists", "label": "Number theorists", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical.scientists.mathematics-statistics.number-theorists" } ] }, { "id": "geo.ind.t1800", "label": "India · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.ind.t1800", "path": [ { "id": "geo.ind", "label": "India", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.ind" }, { "id": "geo.ind.t1800", "label": "India · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.ind.t1800" } ] } ], "excerpt": "Sarvadaman Chowla (1907–1995) was an Indian number theorist who published more than 350 papers over sixty-two years, including the Chowla–Selberg formula and the still-open Chowla conjecture.", "snippet": "Sarvadaman Chowla (1907–1995) was an Indian number theorist who published more than 350 papers over sixty-two years, including the Chowla–Selberg formula and the still-open Chowla conjecture.", "node": "physical.scientists.mathematics-statistics.number-theorists.analytic-number-theorists", "markdown": "# Sarvadaman Chowla\n\n**Sarvadaman Chowla** (22 October 1907 – 10 December 1995) was a number theorist, born in London to Indian parents and educated in Lahore, who wrote more than 350 papers over sixty-two years and left his name on theorems, formulas, and conjectures across analytic and additive number theory, including the Chowla–Selberg formula, the Ankeny–Artin–Chowla theorem, the Chowla–Mordell theorem, and the still-open Chowla conjecture on the [Liouville function](https://www.edgepedia.org/liouville-function).[1](https://www.ams.org/notices/199805/comm-chowla.pdf) • [2](https://mathshistory.st-andrews.ac.uk/Biographies/Chowla/) His associates called him the \"poet of mathematics,\" and he is counted among the best-known Indian number theorists in the tradition of Ramanujan.[1](https://www.ams.org/notices/199805/comm-chowla.pdf)\n\n| Key fact | Detail |\n|---|---|\n| Born / died | 22 October 1907, London; 10 December 1995, Laramie, Wyoming[1](https://www.ams.org/notices/199805/comm-chowla.pdf) |\n| Training | MA, Government College, Lahore, 1928; PhD, Trinity College, Cambridge, 1931, under J. E. Littlewood[1](https://www.ams.org/notices/199805/comm-chowla.pdf) • [3](https://www.mathgenealogy.org/id.php?id=12369) |\n| Output | About 350 papers from 1925 to 1986, spanning analytic number theory, binary quadratic forms, combinatorics, and exponential sums[2](https://mathshistory.st-andrews.ac.uk/Biographies/Chowla/) |\n| Named results | Chowla–Selberg formula (1967), Ankeny–Artin–Chowla theorem (1952), Chowla–Mordell theorem (1962), Bruck–Chowla–Ryser theorem, Mian–Chowla conjecture[1](https://www.ams.org/notices/199805/comm-chowla.pdf) • [4](https://bhavana.org.in/sarvadaman-chowla-the-perpetual-ambassador-for-number-theory/) • [5](https://www.tribuneindia.com/news/spectrum/100-years-of-punjab-school-of-mathematics/) |\n| Chowla conjecture | 1965 conjecture that correlations of the Liouville function vanish; unproven for any k ≥ 2[6](https://msp.org/ant/2015/9-9/ant-v9-n9-p04-s.pdf) |\n| Career after 1947 | Fled Lahore at Partition; IAS Princeton, University of Kansas, University of Colorado, Penn State research professor 1963–1976[1](https://www.ams.org/notices/199805/comm-chowla.pdf) |\n| Primary record | *The Collected Papers of Sarvadaman Chowla*, three chronological volumes (1999), with letters from Hardy, Weil, Selberg, Erdős, and others[7](https://www3.canisius.edu/~huard/chowla.html) |\n\n## Life and career\n\nChowla was born in London on 22 October 1907, the son of Gopal and Shankuntala Chowla, and was educated in Lahore, receiving a master's degree from Government College in 1928.[1](https://www.ams.org/notices/199805/comm-chowla.pdf) Between 1929 and 1931 he studied at [Trinity College, Cambridge](https://www.edgepedia.org/trinity-college-cambridge), taking his doctorate under J. E. Littlewood with a dissertation titled *Contributions to the Analytic Theory of Numbers*.[1](https://www.ams.org/notices/199805/comm-chowla.pdf) • [3](https://www.mathgenealogy.org/id.php?id=12369) His father died of pneumonia in Paris in December 1929 while Chowla was studying abroad.[4](https://bhavana.org.in/sarvadaman-chowla-the-perpetual-ambassador-for-number-theory/)\n\n**Lahore and Partition.** From 1936 to 1947 Chowla headed the Department of Mathematics at Government College of Punjab University in Lahore, where his students included Faqir Chand Auluck, Ram Prakash Bambah, and [Abdus Salam](https://www.edgepedia.org/abdus-salam), the future physics Nobel laureate.[1](https://www.ams.org/notices/199805/comm-chowla.pdf) • [4](https://bhavana.org.in/sarvadaman-chowla-the-perpetual-ambassador-for-number-theory/) The Indian Independence Act of July 1947 placed Lahore in Pakistan, close to the new border; Chowla fled with his family, first to Delhi.[2](https://mathshistory.st-andrews.ac.uk/Biographies/Chowla/) After unsuccessful applications for positions in India, the family moved to the United States in early 1948 on a temporary membership at the [Institute for Advanced Study](https://www.edgepedia.org/institute-for-advanced-study) in Princeton; his escape from Lahore was likely arranged by his wife Himani Mozoomdar, whom he had married while at [St. Stephen's College, Delhi](https://www.edgepedia.org/st-stephens-college-delhi).[4](https://bhavana.org.in/sarvadaman-chowla-the-perpetual-ambassador-for-number-theory/) • [1](https://www.ams.org/notices/199805/comm-chowla.pdf)\n\n**American career.** He visited the IAS until the fall of 1949, then held positions at the [University of Kansas](https://www.edgepedia.org/university-of-kansas) (from 1949) and the University of Colorado (from 1952), before accepting a research professorship at [Pennsylvania State University](https://www.edgepedia.org/pennsylvania-state-university) in 1963, where he remained until his retirement in 1976.[1](https://www.ams.org/notices/199805/comm-chowla.pdf) He died in [Laramie, Wyoming](https://www.edgepedia.org/laramie-wyoming), on 10 December 1995, at the age of 88.[1](https://www.ams.org/notices/199805/comm-chowla.pdf) • [8](https://www.ias.ac.in/article/fulltext/reso/017/09/0855-0883)\n\n## Major mathematical contributions\n\n**The Chowla–Selberg formula.** In 1967 Chowla and [Atle Selberg](https://www.edgepedia.org/atle-selberg) published a formula, now known as the Chowla–Selberg formula, that gives explicitly the product of \\( |\\eta((b+\\sqrt{d})/2a)| \\) as \\( [a,b,c] \\) runs through the classes of \\( H(d) \\) for a fundamental discriminant \\( d \\), where \\( \\eta \\) is the Dedekind eta function.[1](https://www.ams.org/notices/199805/comm-chowla.pdf) The collaboration began at the IAS with Chowla's question about the quadratic L-function of \\( \\mathbb{Q}(\\sqrt{-163}) \\) at \\( s = 1/2 \\); the two announced their results in the *Proceedings of the National Academy of Sciences* in 1949, but the complete article with proofs appeared only in 1967 in Crelle's Journal, and it remains Selberg's only co-authored publication.[4](https://bhavana.org.in/sarvadaman-chowla-the-perpetual-ambassador-for-number-theory/)\n\n**Class numbers and Gauss sums.** The Ankeny–Artin–Chowla theorem on the class number of real quadratic fields appeared in the *Annals of Mathematics* in 1952; it grew out of Chowla's meeting at the IAS with Ankeny, a student of [Emil Artin](https://www.edgepedia.org/emil-artin).[4](https://bhavana.org.in/sarvadaman-chowla-the-perpetual-ambassador-for-number-theory/) The Chowla–Mordell theorem on Gauss sums, proved independently by Chowla and by L. J. Mordell in 1962, states that \\( \\varepsilon(\\chi) \\) is a root of unity if and only if \\( \\chi(n) \\) is the [Legendre symbol](https://www.edgepedia.org/legendre-symbol) \\( (n/p) \\).[1](https://www.ams.org/notices/199805/comm-chowla.pdf) Chowla's name also attaches to the Bruck–Chowla–Ryser theorem on block designs in combinatorics.[1](https://www.ams.org/notices/199805/comm-chowla.pdf)\n\n**Early work.** His papers from 1926 to 1935 cover [Waring's problem](https://www.edgepedia.org/warings-problem), Hypothesis K of Hardy and Littlewood, Ramanujan's conjecture on the numerators of [Bernoulli numbers](https://www.edgepedia.org/bernoulli-numbers), the least prime in an arithmetic progression, and Heilbronn's class number theorem.[9](https://portal.mardi4nfdi.de/wiki/Item:Q2726363) With S. S. Pillai in 1931 he showed that the period \\( l \\) of the continued fraction of \\( \\sqrt{N} \\) satisfies bounds of order \\( \\sqrt{N} \\) for infinitely many \\( N \\) and is on average of order \\( \\sqrt{N} \\).[1](https://www.ams.org/notices/199805/comm-chowla.pdf)\n\n## The Chowla conjecture\n\nIn 1965 Chowla stated the conjecture that now carries his name. For any distinct natural numbers \\( h_1, \\ldots, h_k \\), it asserts that\n\n\\[ \\sum_{n \\le X} \\lambda(n+h_1) \\cdots \\lambda(n+h_k) = o(X) \\quad \\text{as } X \\to \\infty, \\]\n\nwhere \\( \\lambda \\) is the Liouville function.[6](https://msp.org/ant/2015/9-9/ant-v9-n9-p04-s.pdf) Versions replacing the Liouville function with the [Möbius function](https://www.edgepedia.org/mobius-function) also circulate in the literature.[10](https://arxiv.org/pdf/2501.10962) The conjecture encodes the expectation that the prime decomposition of an integer \\( n \\) should be independent of that of \\( n+h \\) for any fixed \\( h \\ge 1 \\), so that the correlations of the sign function vanish.[10](https://arxiv.org/pdf/2501.10962)\n\nThe conjecture remains open for any \\( k \\ge 2 \\), and it is believed to be at least as deep as the twin prime conjecture.[6](https://msp.org/ant/2015/9-9/ant-v9-n9-p04-s.pdf) • [11](https://annals.math.princeton.edu/wp-content/uploads/annals-v183-n3-p06-p.pdf)\n\n## Collaborations and influence\n\nChowla worked with a wide circle. The collected-works bibliography lists nearly 70 co-authors, while the MacTutor biography names around 40, including T. M. Apostol, Emil Artin, Richard Brauer, Harold Davenport, Paul Erdős, Marshall Hall, Helmut Hasse, L. J. Mordell, S. S. Pillai, [C. R. Rao](https://www.edgepedia.org/c-r-rao), Atle Selberg, Goro Shimura, Thoralf Skolem, and Hans Zassenhaus.[9](https://portal.mardi4nfdi.de/wiki/Item:Q2726363) • [2](https://mathshistory.st-andrews.ac.uk/Biographies/Chowla/)\n\n**Erdős and Kansas.** During his Kansas years, 1949 to 1952, Chowla and Erdős wrote two papers, one with Paul Bateman, on special values of Dirichlet L-functions.[4](https://bhavana.org.in/sarvadaman-chowla-the-perpetual-ambassador-for-number-theory/) Erdős had engaged Chowla's problems much earlier: his 1936 paper \"On a problem of Chowla and some related problems\" treats Chowla's conjecture on the divisor function \\( d(m) \\) of consecutive integers.[12](https://www.renyi.hu/~p_erdos/1936-03.pdf)\n\n**Pillai.** Chowla and S. S. Pillai (1901–1950), described together as two of the foremost Indian mathematicians of the generation after Ramanujan, maintained a regular correspondence from the 1920s until one month before Pillai's death in 1950, covering Waring's problem, binary quadratic forms, class numbers, and error asymptotics for Euler's function.[8](https://www.ias.ac.in/article/fulltext/reso/017/09/0855-0883)\n\n**Students.** A 1979 special issue of the *Journal of Number Theory* listed twenty-three doctoral students of Chowla in the United States.[1](https://www.ams.org/notices/199805/comm-chowla.pdf) Other counts differ: he supervised 13 students at the University of Colorado at Boulder, and press summaries give 25 PhD students in total.[4](https://bhavana.org.in/sarvadaman-chowla-the-perpetual-ambassador-for-number-theory/) • [5](https://www.tribuneindia.com/news/spectrum/100-years-of-punjab-school-of-mathematics/) His doctoral students include W. E. Mientka (1955), J. B. Friedlander, J. G. Huard (1978), and his daughter Paromita, born 1934, who also became a number theorist and professor at Pennsylvania State University.[9](https://portal.mardi4nfdi.de/wiki/Item:Q2726363) • [1](https://www.ams.org/notices/199805/comm-chowla.pdf)\n\n## Recent developments\n\nWork on Chowla's conjecture has continued to advance on several fronts, though the full conjecture remains open.\n\n**Averaged and logarithmic forms.** Matomäki, Radziwiłł, and Tao proved in 2015 an averaged form of the conjecture: summing the products \\( \\lambda(n+h_1) \\cdots \\lambda(n+h_k) \\) over all shifts \\( h_1, \\ldots, h_k \\le H \\) gives \\( o(H^k X) \\) for \\( 10 \\le H \\le X \\), using mean values of multiplicative functions in short intervals together with an argument of Kátai and Bourgain–Sarnak–Ziegler.[6](https://msp.org/ant/2015/9-9/ant-v9-n9-p04-s.pdf) Earlier partial results include the binary case \\( h_1 = 0 \\), \\( h_2 = 1 \\) by Harman, Pintz, and Wolke in 1985, and a \\( \\delta(h) > 0 \\) bound for any single shift by Matomäki and Radziwiłł in 2016.[13](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/chowla-conjecture-and-landausiegel-zeroes/515F3378450DED201A21E26BF801CEEB) The logarithmically averaged Chowla conjecture has been proven in stages, for odd \\( k \\) and for \\( k = 2 \\), in works of Tao, Tao and Teräväinen, Helfgott and Radziwiłł, and Pilatte.[13](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/chowla-conjecture-and-landausiegel-zeroes/515F3378450DED201A21E26BF801CEEB)\n\n**Conditional results.** A 2024/2025 paper in the *Mathematical Proceedings of the Cambridge Philosophical Society* establishes a non-trivial bound for the sums \\( \\sum_{n \\le x} \\lambda(n+h_1) \\cdots \\lambda(n+h_k) \\) under the assumption that a Landau–Siegel zero exists, improving on earlier work of Germán–Kátai, Chinis–Tao, and Teräväinen.[13](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/chowla-conjecture-and-landausiegel-zeroes/515F3378450DED201A21E26BF801CEEB) A January 2025 preprint surveys variants of the conjecture, including the Möbius-function versions.[10](https://arxiv.org/pdf/2501.10962)\n\n**Other Chowla problems.** Research on Chowla's problem on arithmetical functions, concerning functions \\( f \\) with \\( f(a) = 0 \\) for \\( 1 < (a,q) < q \\), has shown that no such function exists when \\( f \\) takes values in an algebraic number field disjoint from the \\( q \\)-th cyclotomic field.[14](https://www.labmath.uqam.ca/~annales/volumes/35-2/PDF/229-237.pdf)\n\n## References\n\n1. [Sarvadaman Chowla 1907–1995, AMS Notices (May 1998)](https://www.ams.org/notices/199805/comm-chowla.pdf)\n2. [Sarvadaman Chowla (1907–1995), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Chowla/)\n3. [Sarvadaman Chowla, Mathematics Genealogy Project](https://www.mathgenealogy.org/id.php?id=12369)\n4. [Sarvadaman Chowla: The Perpetual Ambassador for Number Theory, Bhāvanā](https://bhavana.org.in/sarvadaman-chowla-the-perpetual-ambassador-for-number-theory/)\n5. [100 years of Punjab School of Mathematics, The Tribune](https://www.tribuneindia.com/news/spectrum/100-years-of-punjab-school-of-mathematics/)\n6. [Matomäki, Radziwiłł, Tao (2015). An averaged form of Chowla's conjecture. Algebra & Number Theory](https://msp.org/ant/2015/9-9/ant-v9-n9-p04-s.pdf)\n7. [The Collected Papers of Sarvadaman Chowla (1999)](https://www3.canisius.edu/~huard/chowla.html)\n8. [Pillai and Chowla, Resonance, Indian Academy of Sciences](https://www.ias.ac.in/article/fulltext/reso/017/09/0855-0883)\n9. [Review of The Collected Papers of Sarvadaman Chowla, Zentralblatt/MaRDI](https://portal.mardi4nfdi.de/wiki/Item:Q2726363)\n10. [On variants of Chowla's conjecture, arXiv:2501.10962 (January 2025)](https://arxiv.org/pdf/2501.10962)\n11. [Multiplicative functions in short intervals, Annals of Mathematics (2016)](https://annals.math.princeton.edu/wp-content/uploads/annals-v183-n3-p06-p.pdf)\n12. [Erdős (1936). On a problem of Chowla and some related problems](https://www.renyi.hu/~p_erdos/1936-03.pdf)\n13. [The Chowla conjecture and Landau–Siegel zeroes, Math. Proc. Camb. Phil. Soc.](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/chowla-conjecture-and-landausiegel-zeroes/515F3378450DED201A21E26BF801CEEB)\n14. [Some Remarks on a Problem of Chowla, Annales des sciences mathématiques du Québec](https://www.labmath.uqam.ca/~annales/volumes/35-2/PDF/229-237.pdf)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Physical and mathematical scientists › Mathematicians and statisticians › Number theorists › Analytic number theorists*\n\n*Initially written Oct 10, 2026 · Edited: Oct 11, 2026; Oct 15, 2026; Oct 21, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://www3.canisius.edu/~huard/chowla.html" ], "url": "https://www.edgepedia.org/sarvadaman-chowla", "markdown_url": "https://www.edgepedia.org/sarvadaman-chowla.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Sarvadaman Chowla\", Edgepedia (EdgeChat), https://www.edgepedia.org/sarvadaman-chowla. Edgepedia Community License 1.0.", "credit_md": "\"[Sarvadaman Chowla](https://www.edgepedia.org/sarvadaman-chowla)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/sarvadaman-chowla](https://www.edgepedia.org/sarvadaman-chowla). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Sarvadaman Chowla\", Edgepedia (EdgeChat), https://www.edgepedia.org/sarvadaman-chowla. Edgepedia Community License 1.0.", "speakable": "Sarvadaman Chowla was an Indian number theorist who published more than 350 papers over sixty-two years, including the Chowla–Selberg formula and the still-open Chowla conjecture." }