The foundation of analytic number theory (1863)

DIRICHLET, Johann Peter Gustav Lejeune [Richard DEDEKIND]

Vorlesungen über Zahlentheorie.

Published: Braunschweig, Friedrich Vieweg und Sohn

Date: 1863

8vo (215 × 150 mm), pp. xiii, [1, blank], 414, [2, blank]; a largely bright, attractive text, light finger soiling, and occasional spotting, especially of initial leaves, one or two very small pencil annotations, light ink spotting to pp. 31-32; very good in modern calf-backed marbled boards, new marbled endpapers

A very fine copy of the first edition. Dirichlet, most famous for his deep contributions to number theory, has, furthermore, to be considered the founder of analytic number theory. ‘One of the major developments in number theory is the introduction of analytic methods and of analytic results to express and prove facts about integers … The first deep and deliberate use of analysis to tackle what seemed to be a clear problem of algebra was made by Peter Gustav Lejeune-Dirichlet (1805-1859), a student of Gauss and Jacobi, professor at Breslau and Berlin, and then successor to Gauss at Göttingen. Dirichlet’s great work, the Vorlesungen über Zahlentheorie, expounded Gauss’s Disquisitiones and gave his own contributions. ‘The problem that caused Dirichlet to employ analysis was to show that every arithmetic sequence a, a + b, a + 2b, a + 3b, … a + nb, …, where a and b are relatively prime, contains an infinite number of primes … The chief problem involving the introduction of analysis concerned the function π(x) which represents the number of primes not exceeding x. Thus π(8) is 4 since 2, 3, 5, and 7 are prime and π(11) is 5. As x increases the additional primes become scarcer and the problem was, What is the proper analytical expression for π(x)?’ (Kline, pp. 829-830) The working through of this, and related problems, makes the Vorlesungen a kind of watershed between the earlier number theory of Fermat, Jacobi and Gauss, and the number theory that was to follow with the work of Dedekind, Riemann, Hilbert and others. The contents of the Vorlesungen über Zahlentheorie, were gathered by Richard Dedekind (1831-1916) from Dirichlet’s lectures on number theory given at Göttingen after 1855 when he took on Gauss’s role there. The five chapters of Dirichlet’s works cover the divisibility of numbers, congruence, quadratic residues, quadratic forms and the determination of the class number of binary quadratic forms. Dedekind’s supplements include additional chapters on related problems. The main findings of the Vorlesungen are the class number formulae for binary quadratic forms and the proof that arithmetic progressions contains an infinite number of primes (Dirichlet’s theorem), which proof introduced Dirichlet L-series, both foundational for analytic number theory. A second edition was published in 1871 and in the 1876 third edition Dedekind added a further two supplements in which he began to develop the theory of ideals.

Very good

Quarter calf