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Publisher:

American Mathematical Society and London Mathematical Society

Publication Date:

2005

Number of Pages:

278

Format:

Hardcover

Series:

History of Mathematics 26

Price:

59.00

ISBN:

0-8218-3457-6

Category:

Monograph

[Reviewed by , on ]

Michael Berg

10/22/2005

Michael Rosen, in his excellent and informative foreword to Robert Burns' recently published translation, *The St. Petersburg School of Number Theory*, of B. N. Delone's 1947 book, *Petersburgskaia Shkola Teorii Chisel*, notes that

…by composing the present volume, Delone did a great service to posterity. The new translation should have a wide readership among English speaking mathematicians with enough background to enjoy it…

Earlier, Rosen says that "[i]t should be emphasized that this book is strictly about number theory," and it might even be proper to say that the book is dominated by analytic methods. Delone himself makes no bones about the primacy of St. Petersburg as far as this discipline is concerned:

Analytic number theory was in large part the creation of the St. Petersburg school. The first important results in this area are found in the works of Euler. The next big step forward in analytic number theory was made by Chebyshev in his ingenious work on the distribution of the primes, serving as the starting point for later work of Riemann and Hadamard…

Quite justly, Burns interjects a translator's footnote in this passage, objecting that Dirichlet should have been mentioned right after Euler. But Delone goes on (p. 212):

Finally, Vinogradov invented a remarkable new arithmetic approach to problems of analytic number theory (the arithmetic of inequalities), allowing him to make extraordinary discoveries constituting some of the glories of Soviet science.

In 1941 Ivan Matveevich (Vinogradov) was awarded the Stalin prize first class for his book, "A new method in the analytic theory of numbers," and in 1945 by decree of the Presidium of the Supreme Soviet of the USSR was accorded the title of Hero of Socialist Labour.

Thus Delone emerges as a man of his time and place, possessing an idiosyncratic writing style (certainly from our Western point of view), which is at times propagandist and at times entertaining. There are, for instance, a number of good stories in the book, interspersed between very serious (and quite informative) commentary on mathematics properly so-called; here is a sample: "Chebyshev almost never missed a lecture, was never late, and at the sound of the bell announcing the end of the lecture, stopped talking immediately, sometimes even in the middle of a word."

Before getting to the book's mathematical content, it is perhaps permissible to add an anecdote about Delone himself, in order to add some background color to the experience of stepping back in time with him as our guide. According to E. M. Landis in "About Mathematics at Moscow State University in the late 1940's and early 1950's," appearing on pp. 55–73 of *Golden Years of Moscow Mathematics* (S. Zdravkovska, P. L. Duren, Eds.), History of Mathematics Series, Vol 6 (AMS, LMS), 1993,

Analytic geometry was taught by B. N. Delone who acted strangely. For instance, when explaining symmetry, he drew an elephant on the black board, stood in front of the board on a chair, and pretended to be an elephant. To produce the right effect he moved his hand in front of his face like an elephant trunk…

Bizarre and memorable, yes, but, to be fair, we must juxtapose this data against the fact that Delone was the advisor of none other than I. R. Shafarevich (loc. cit.). And, indeed, the analysis and evaluation Delone supplies *vis à vis* the number theoretic works of the mathematicians of the St. Petersburg School is on the money.

The mathematicians featured in *The St. Petersburg School of Number Theory* are Chebyshev, Korkin, Zolotarev, Markov (*fils* not *père*), Voronoi, and Vinogradov. Delone presents compact discussions of these scholars' major arithmetical contributions, including Chebyshev's work on π(x) (the number of primes ≤ x), material on quadratic forms by Korkin, Markov, and Voronoi, work by Zolotarev on ideals and on theories of divisors in orders of number fields, and Vinogradov's well-known contributions to Waring's Problem and to Goldbach's Conjecture. We encounter a marvelous cross-section of topics from 19-th and early 20-th century number theory, very classical themes today, and certainly possessed of a classical beauty. It is exciting to look at these familiar themes from the unusual perspective Delone affords, coming from an earlier time and another culture, or, rather, a pair of such: the St. Petersburg School thrived during both the Czarist and communist eras of Russia and Delone's remarks give witness to this schizoid reality. It's a very interesting book on a number of counts.

Michael Berg is Professor of Mathematics at Loyola Marymount University.

*Pafnutii L'vovich Chebyshev (1821-1894)*

- Pafnutii L'vovich Chebyshev
- Chebyshev's articles on the prime numbers

*Aleksandr Nikolaevich Korkin (1837-1908)*

- Aleksandr Nikolaevich Korkin
- The articles of Korkin and Zolotarev on the minima of positive quadratic forms

*Egor Ivanovich Zolotarev (1847-1878)*

- Egor Ivanovich Zolotarev
- Zolotarev's memoirs on the theory of ideal numbers

*Andrei Andreevich Markov (1856-1922)*

- Andrei Andreevich Markov
- On binary quadratic forms of positive determinant

*Georgii Fedoseevich Voronoi (1868-1908)*

- Georgii Fedoseevich Voronoi
- Voronoi's dissertations on algebraic numbers of the third degree
- Voronoi's memoir of 1903: "On a problem from the theory of asymptotic functions"
- Voronoi's memoirs on quadratic forms

*Ivan Matveevich Vinogradov (1891-1983)*

- Ivan Matveevich Vinogradov
- Works of Vinogradov from the first period of his mathematical activity
- Waring's problem
- The Goldbach problem
- Estimation of Weyl sums and the problem of the fractional parts of a polynomial
- Bibliography

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