Multiplicative Number Theory I. Classical Theory

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theorem_3_9: Theorem 3.9 and its historical note give the error-free upper bound 2y/(phi(q) log(y/q)) for primes in a reduced residue class in an interval of length y>=2q; only that domain and the statement are extracted.

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Hugh L. Montgomery and Robert C. Vaughan, Multiplicative Number Theory I. Classical Theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, 2006, ISBN 9780521849036, book DOI.

The selected artifact is the publisher's 32-page Chapter 3, Principles and first examples of sieve methods, printed pp. 76–107, served on Vaughan's author page. Its physical page 1 is printed p. 76; printed pp. 90 and 102 are physical pages 15 and 27. The chapter carries the Cambridge Books Online copyright notice of 2010. The author-hosted frontmatter identifies the print publication as 2006; the electronic publication date is distinct from that print year. The whole book is not the selected artifact.

Provenance: the author-hosted chapter, chapter DOI 10.1017/CBO9780511618314.005.

The consumed result is Theorem 3.9 with the Section 3.5 notes on Section 3.3: for a reduced residue class modulo $q$, the number of primes in an interval of length $y\ge2q$ is at most $2y/(\varphi(q)\log(y/q))$. The theorem on p. 90 first gives an error factor; the notes on p. 102 explicitly state that the error can be omitted, crediting Montgomery and Vaughan's 1973 paper. This is a primary-author textbook statement of that sharpening, not an examination of the 1973 proof.

Read status. Claims checked for Theorem 3.9 and the cited sharpening, including the coprimality condition and the domain $x\ge0$, $y\ge2q$; both printed pages were inspected. No source proof is verified here, and no claim about the additional range $q<y<2q$ is extracted from these pages.

Bears on. Problem 12(iii): the inequality supplies an upper bound for prime reciprocal sums in reduced classes, relevant to the normalization route. It does not supply that route's missing multiscale harmonic-mass estimate.

No file of this source is held: no license on record permits its redistribution. The chapter footer reads “© Cambridge University Press, 2010”; the frontmatter reads “© Cambridge University Press 2006” and requires permission for reproduction beyond its stated exceptions. Those notices were read on 2026-10-16; no open license is stated there.