Book Review
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MathSciNet review:
3020833
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Book Information:
Authors:
Alina Carmen Cojocaru and
M. Ram Murty
Title:
An introduction to sieve methods and their applications
Additional book information:
London Mathematical Society Student Texts, 66,
Cambridge University Press,
Cambridge,
2006,
xii+224 pp.,
ISBN 978-0-521-64275-3
Authors:
John Friedlander and
Henryk Iwaniec
Title:
Opera de cribro
Additional book information:
American Mathematical Society Colloquium Publications, 57,
American Mathematical Society,
Providence, RI,
2010,
xx+527 pp.,
ISBN 978-0-8218-4970-5
Carter Bays and Richard H. Hudson, A new bound for the smallest $x$ with $\pi (x)>\textrm {li}(x)$, Math. Comp. 69 (2000), no. 231, 1285–1296. MR 1752093, DOI 10.1090/S0025-5718-99-01104-7
Harold Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR 1790423
Harold G. Diamond and H. Halberstam, A higher-dimensional sieve method, Cambridge Tracts in Mathematics, vol. 177, Cambridge University Press, Cambridge, 2008. With an appendix (“Procedures for computing sieve functions”) by William F. Galway. MR 2458547, DOI 10.1017/CBO9780511542909
John Friedlander and Henryk Iwaniec, The polynomial $X^2+Y^4$ captures its primes, Ann. of Math. (2) 148 (1998), no. 3, 945–1040. MR 1670065, DOI 10.2307/121034
Daniel A. Goldston, János Pintz, and Cem Y. Yıldırım, Primes in tuples. I, Ann. of Math. (2) 170 (2009), no. 2, 819–862. MR 2552109, DOI 10.4007/annals.2009.170.819
A. Granville and K. Soundararajan, Multiplicative number theory, book in preparation.
George Greaves, Sieves in number theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 43, Springer-Verlag, Berlin, 2001. MR 1836967, DOI 10.1007/978-3-662-04658-6
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), no. 2, 481–547. MR 2415379, DOI 10.4007/annals.2008.167.481
H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1974. MR 0424730
Glyn Harman, Prime-detecting sieves, London Mathematical Society Monographs Series, vol. 33, Princeton University Press, Princeton, NJ, 2007. MR 2331072
D. R. Heath-Brown, Prime twins and Siegel zeros, Proc. London Math. Soc. (3) 47 (1983), no. 2, 193–224. MR 703977, DOI 10.1112/plms/s3-47.2.193
D. R. Heath-Brown, Primes represented by $x^3+2y^3$, Acta Math. 186 (2001), no. 1, 1–84. MR 1828372, DOI 10.1007/BF02392715
Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
D. Koukoulopoulos, Pretentious multiplicative functions and the prime number theorem for arithmetic progressions, preprint, http://arxiv.org/abs/1203.0596
Hugh L. Montgomery and Robert C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR 2378655
References
- C. Bays and R.H. Hudson, A new bound for the smallest x with $\pi (x) > \mathrm {li}(x)$, Math. Comp. 69 (1999), no. 231, 1285–1296. MR 1752093 (2001c:11138)
- H. Davenport (as revised by H. Montgomery), Multiplicative number theory, Springer-Verlag, New York, 2000. MR 1790423 (2001f:11001)
- H. Diamond and H. Halberstam, with an appendix by W. Galway, A higher-dimensional sieve method, Cambridge University Press, Cambridge, 2008. MR 2458547 (2009h:11151)
- J. Friedlander and H. Iwaniec, The polynomial $X^2+Y^4$ captures its primes, Ann. of Math. (2) 148 (1998), no. 3, 945–1040. MR 1670065 (2000c:11150a)
- D. Goldston, J. Pintz, and C. Y. Yıldırım, Primes in tuples. I, Ann. of Math. (2) 170 (2009), no. 2, 819–862. MR 2552109 (2011c:11146)
- A. Granville and K. Soundararajan, Multiplicative number theory, book in preparation.
- G. Greaves, Sieves in number theory, Springer-Verlag, Berlin, 2001. MR 1836967 (2002i:11092)
- B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), no. 2, 481–547. MR 2415379 (2009e:11181)
- H. Halberstam and H.-E. Richert, Sieve methods, Dover, 2011. (Reprint of Academic Press 1974 edition.) MR 0424730 (54:12689)
- G. Harman, Prime-detecting sieves, Princeton University Press, Princeton, 2007. MR 2331072 (2008k:11097)
- D. R. Heath-Brown, Prime twins and Siegel zeros, Proc. London Math. Soc. (3) 47 (1983), no. 2, 193–224. MR 703977 (84m:10029)
- D. R. Heath-Brown, Primes represented by $x^3 + 2y^3$, Acta Math. 186 (2001), no. 1, 1–84. MR 1828372 (2002b:11122)
- H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society, Providence, 2004. MR 2061214 (2005h:11005)
- D. Koukoulopoulos, Pretentious multiplicative functions and the prime number theorem for arithmetic progressions, preprint, http://arxiv.org/abs/1203.0596
- H. Montgomery and R. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge University Press, Cambridge, 2007. MR 2378655 (2009b:11001)
Review Information:
Reviewer:
Frank Thorne
Affiliation:
Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, South Carolina 29208
Email:
thorne@math.sc.edu
Journal:
Bull. Amer. Math. Soc.
50 (2013), 359-366
DOI:
https://doi.org/10.1090/S0273-0979-2012-01390-3
Published electronically:
October 5, 2012
Review copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.