The collected papers of Sarvadaman Chowla. Vol. I: 1925--1935. Edited by James G. Huard and Kenneth S. Williams (Q2726363)

This is a joint review for Vols. I--III (for bibliographical details see Zbl 1036.01016 and Zbl 1036.01017).NEWLINENEWLINESarvadaman Chowla, born in London 1907, worked in the field of number theory, first in India, then in Princeton, Kansas, Colorado, and Pennsylvania. He died in 1995. Between 1925 and 1986 Chowla wrote more than 350 mathematical papers. Number theorists are grateful to the editors, J. G. Huard and K. S. Williams, for making the extensive number theoretical work of S. Chowla easily accessible in a three volume set. Moreover, these collected works contain a description of Chowla's life by J. G. Huard, a short description (19 pages) of the work of Chowla by Ram Murty, Kumar Murty \& K. S. Williams. They contain \textit{Memories of Chowla}, written by his daughter Paromita (also working in number theory), \textit{Recollections of Chowla}\ by R. Ayoub, C. R. Rao, R. P. Bambah, P. T. Bateman, A. Selberg, T. M. Apostol and many others, reproductions of letters written to Chowla by G. Hardy, S. Chandrasekhar, P. Turán, A. Selberg, A. Weil, J.-P. Serre, and P. Erdős. The bibliography of S. Chowla (with 34 solutions of problems from the Journal of the Indian Mathematical Society [reproduced in Vol. I, p. 1--56], and 351 papers from 1926 to 1986, with a maximum of 25 papers in the year 1934) is given. A list of the Ph.D. students of S. Chowla is collected, beginning with W. E. Mientka (1955), and including, for example, his daughter Paromita, H. Walum, S. L. Segal, A. M. Vaidya, R. A. Smith, K. Thanigasalam, J. B. Friedlander, and others, and ending with J. G. Huard (1978), one of the editors. There is also a list of the co-authors of Chowla (nearly 70 in number), and an index of the subjects of Chowla's papers.NEWLINENEWLINEVolume I contains Chowla's mathematical papers from 1926--1935, the numbers 1 -- 93 from the bibliography [there are many short papers]. We mention some of the topics treated in the collected papers, reproduced in volume I:NEWLINENEWLINENew proof for Gauss' formula \(\sum_{d\,| \,n} \varphi(d) =n\), a new proof for the estimate \(d(n) = {\mathcal O}(n^{\varepsilon})\), sums of some infinite series, series expansion for \(K(k) = \int_0^{{1\over 2}\pi} {{dt}\over {\sqrt{1-k^2\sin^2 t}}}\), asymptotics for \(\sum_{n\leq x} \varphi^r(n)\), investigation of the remainder term in \(\sum_{n\leq x} {{\varphi(n)}\over n} - {6\over {\pi^2}} x\), lower estimate of the greatest prime factor of \(\prod_{n\leq r} (1+n^2)\) and of \((x^2+1)\), Waring's problem for cubes, the easier Waring problem, zeta formulae, Hypothesis \(K\) of Hardy and Littlewood, continued fractions, lattice points in ellipsoids, estimates for \(\sum_{n\leq x} f(n) \cdot e^{2\pi i \theta n}\), conjecture of Ramanujan on the numerators of Bernoulli numbers, primes, the least prime in an arithmetic progression, \(\Omega\)-estimate of \(\sum \chi(n)\), class number of binary quadratic forms, Heilbronn's class number theorem, and others.NEWLINENEWLINEVolume II contains Chowla's mathematical papers from 1936--1961, the numbers 94--230 from the bibliography. Topics treated in papers of Chowla in this volume are, for example:NEWLINENEWLINEWarings's problem (Hypothesis \(K\), exact formulae for \(g(n)\), \(\Omega\)-estimate of \(r_{n,n}(N)\), the number of solutions of the diophantine equation \(\left\{ N=s_1^n+\dots+s_n^n\right\}\), representation as sums of squares, Waring's problem for cubes, \dots), size of \(L(1,\chi)\), values of \(L\)-functions, character sums, the partition function, a differential equation for \(\sum_1^\infty p(n) x^n\), zeta-functions (including the power series coefficients of \(\zeta(s)\)), de Bruijn's functions \(\#\{n\leq x, p_{\max}(n) \leq y\}\) and \(\#\{n\leq x, p_{\max}(n) > y\}\), class numbers of [real quadratic, quadratic, cyclotomic] number fields, lattice points, biquadratic residues, cubic character of \(2 \bmod p\), polynomials modulo \(p\), Diophantine equations, Pell's equation, elementary geometry, Weierstrass' approximation theorem, combinatorial problems, block designs, difference sets, Fourier series, \dotsNEWLINENEWLINE Volume III contains Chowla's mathematical papers from 1962--1986, the numbers 236--351 from the bibliography. Topics treated in these papers are, for example, class numbers of quadratic fields, class numbers of imaginary quadratic fields, class numbers of function fields over \({\mathbb Z}/p{\mathbb Z}\), imaginary quadratic fields with class number one, class-invariants of \({\mathbb Q}(\sqrt{-p})\), fundamental units in real quadratic fields, series connected with Riemann's zeta-function, \(L\)-series, Hecke \(L\)-series, \(L\)-series and elliptic curves, Epstein's zeta-function, functional equation for \( L(s,\chi, Q) = \sum {{\chi(Q(x,y))}\over {Q(x,y)^s}}\), where \(Q(x,y)=x^2+ gy^2\), \(g>0\), Riemann's hypothesis and Hilbert's tenth problem (12 years before the solution of Hilbert's problem by Matiyasevich), Diophantine equations (in quadratic number fields, in cyclotomic fields), Gaussian sums, character sums, Kloosterman's sums, Dedekind sums, congruence properties of Apèry numbers, series for \(\zeta(3)\), algebraic differential equations for some elliptic functions, least quadratic residue modulo \(p\), Bernoulli numbers, and problem-collections.NEWLINENEWLINENumber theorists have to thank the editors for providing them with an easy, rapid access to many topics in number theory, in which S. Chowla was interested and to which he contributed.