# Sarvadaman Chowla
**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)
| Key fact | Detail |
|---|---|
| Born / died | 22 October 1907, London; 10 December 1995, Laramie, Wyoming[1](https://www.ams.org/notices/199805/comm-chowla.pdf) |
| 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) |
| 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/) |
| 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/) |
| 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) |
| 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) |
| 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) |
## Life and career
Chowla 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/)
**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)
**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)
## Major mathematical contributions
**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/)
**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)
**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)
## The Chowla conjecture
In 1965 Chowla stated the conjecture that now carries his name. For any distinct natural numbers \( h_1, \ldots, h_k \), it asserts that
\[ \sum_{n \le X} \lambda(n+h_1) \cdots \lambda(n+h_k) = o(X) \quad \text{as } X \to \infty, \]
where \( \lambda \) 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)
The 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)
## Collaborations and influence
Chowla 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/)
**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)
**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)
**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)
## Recent developments
Work on Chowla's conjecture has continued to advance on several fronts, though the full conjecture remains open.
**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)
**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)
**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)
## References
1. [Sarvadaman Chowla 1907–1995, AMS Notices (May 1998)](https://www.ams.org/notices/199805/comm-chowla.pdf)
2. [Sarvadaman Chowla (1907–1995), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Chowla/)
3. [Sarvadaman Chowla, Mathematics Genealogy Project](https://www.mathgenealogy.org/id.php?id=12369)
4. [Sarvadaman Chowla: The Perpetual Ambassador for Number Theory, Bhāvanā](https://bhavana.org.in/sarvadaman-chowla-the-perpetual-ambassador-for-number-theory/)
5. [100 years of Punjab School of Mathematics, The Tribune](https://www.tribuneindia.com/news/spectrum/100-years-of-punjab-school-of-mathematics/)
6. [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)
7. [The Collected Papers of Sarvadaman Chowla (1999)](https://www3.canisius.edu/~huard/chowla.html)
8. [Pillai and Chowla, Resonance, Indian Academy of Sciences](https://www.ias.ac.in/article/fulltext/reso/017/09/0855-0883)
9. [Review of The Collected Papers of Sarvadaman Chowla, Zentralblatt/MaRDI](https://portal.mardi4nfdi.de/wiki/Item:Q2726363)
10. [On variants of Chowla's conjecture, arXiv:2501.10962 (January 2025)](https://arxiv.org/pdf/2501.10962)
11. [Multiplicative functions in short intervals, Annals of Mathematics (2016)](https://annals.math.princeton.edu/wp-content/uploads/annals-v183-n3-p06-p.pdf)
12. [Erdős (1936). On a problem of Chowla and some related problems](https://www.renyi.hu/~p_erdos/1936-03.pdf)
13. [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)
14. [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)
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*Topic: Encyclopedia › Physical world and mathematics › Physical and mathematical scientists › Mathematicians and statisticians › Number theorists › Analytic number theorists*
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