Book Review

Delone actually gives a simpler argument than Vinogradov’s, proving the less accurate estimate GðnÞ < 6n ln nþ 11. When judged as a work of history this book has certain weaknesses. As mentioned above, Delone repeatedly gives a geometric interpretation of the work of Korkin and Zolotarev, Markov, and the early work of Vorono ı in quadratic forms. But there appears to be little historical evidence to support this interpretation. Delone’s claims may in fact be true, but they are not documented. Also, the reader needs to be aware of a certain bias in the writing. The works of Russian mathematicians appear to tower over the landscape, with the efforts of Gauss, Dirichlet, Dedekind, Hermite, and Minkowski appearing as distant flashes of lightning on the horizon. For example, in reference to Vinogradov’s research on Waring’s problem, Delone states (p. 227): “His method [compared to that of Hardy–Littlewood] may therefore be regarded as coming fully within the tradition of the St. Petersburg school whereby deep results are obtained by simple methods.” Nevertheless, these qualifications notwithstanding, this work is an able discussion of some fascinating number theory.