08-17-2026, 02:21 PM
Multiplicative Number Theory
Author: Harold Davenport
Editor: Hugh L. Montgomery
First publication: 1967
Edition linked on Goodreads: 2nd edition, 1980
Publisher: Springer-Verlag
Series:Graduate Texts in Mathematics, Vol. 74
Field: Analytic / Multiplicative Number Theory
Multiplicative Number Theory is one of the classic texts of analytic number theory, centered on one of the subject’s fundamental questions: how are prime numbers distributed, particularly among arithmetic progressions? Davenport develops the theory from Dirichlet’s theorem on primes in arithmetic progressions and gradually introduces Dirichlet characters, Gauss sums, cyclotomy, class-number formulas, the Riemann zeta function, and Dirichlet $L$-functions. A major strength is the way these subjects are connected rather than treated as isolated theorems: characters and $L$-functions become tools for translating arithmetic questions about primes into questions about complex functions and their zeros. The book develops the functional equations of $L$-functions, zero-free regions for $\zeta(s)$ and $L(s,\chi)$, explicit formulas, and ultimately the Prime Number Theorem and its analogue for arithmetic progressions.
The later part moves toward deeper results and techniques, including Siegel’s theorem, the Pólya–Vinogradov inequality, exponential sums, the large sieve, and Bombieri’s theorem on the average distribution of primes in arithmetic progressions. The second edition, edited by Hugh Montgomery after Davenport's death, substantially revised the treatment of these later topics, notably using Vaughan's approach to Bombieri's theorem. This is therefore not primarily an elementary introduction to number theory; it is a compact, proof-oriented graduate text that leads the reader surprisingly quickly from classical results to methods close to research-level analytic number theory. Springer describes the later third edition as suitable for graduate students and researchers, while the Goodreads listing similarly emphasizes its treatment of Dirichlet's theorem, Siegel's theorem, and the large sieve.
Key takeaways
Springer — Multiplicative Number Theory
Author: Harold Davenport
Editor: Hugh L. Montgomery
First publication: 1967
Edition linked on Goodreads: 2nd edition, 1980
Publisher: Springer-Verlag
Series:Graduate Texts in Mathematics, Vol. 74
Field: Analytic / Multiplicative Number Theory
Multiplicative Number Theory is one of the classic texts of analytic number theory, centered on one of the subject’s fundamental questions: how are prime numbers distributed, particularly among arithmetic progressions? Davenport develops the theory from Dirichlet’s theorem on primes in arithmetic progressions and gradually introduces Dirichlet characters, Gauss sums, cyclotomy, class-number formulas, the Riemann zeta function, and Dirichlet $L$-functions. A major strength is the way these subjects are connected rather than treated as isolated theorems: characters and $L$-functions become tools for translating arithmetic questions about primes into questions about complex functions and their zeros. The book develops the functional equations of $L$-functions, zero-free regions for $\zeta(s)$ and $L(s,\chi)$, explicit formulas, and ultimately the Prime Number Theorem and its analogue for arithmetic progressions.
The later part moves toward deeper results and techniques, including Siegel’s theorem, the Pólya–Vinogradov inequality, exponential sums, the large sieve, and Bombieri’s theorem on the average distribution of primes in arithmetic progressions. The second edition, edited by Hugh Montgomery after Davenport's death, substantially revised the treatment of these later topics, notably using Vaughan's approach to Bombieri's theorem. This is therefore not primarily an elementary introduction to number theory; it is a compact, proof-oriented graduate text that leads the reader surprisingly quickly from classical results to methods close to research-level analytic number theory. Springer describes the later third edition as suitable for graduate students and researchers, while the Goodreads listing similarly emphasizes its treatment of Dirichlet's theorem, Siegel's theorem, and the large sieve.
Key takeaways
- Prime distribution is the central theme: especially primes satisfying $p\equiv a\pmod q$.
- Dirichlet characters and $L$-functions are fundamental tools: they provide the bridge between arithmetic and complex analysis.
- Zeros matter enormously: information about zeros of $\zeta(s)$ and $L(s,\chi)$ translates into information about the distribution of primes.
- The book progresses toward powerful modern methods: particularly the large sieve and Bombieri's theorem, making it an important bridge between classical analytic number theory and research-level techniques.
Springer — Multiplicative Number Theory
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