Prime-detecting sieves

Prime-Detecting Sieves

Volumn , Issue , 2012, Pages 1-362

  Harman, Glyn ^a  

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References (168)

1. L. M. Adleman and D. R. Heath-Brown. The first case of Fermat's last theorem, Invent. Math. 79 (1985), 409-416.
2. N. C. Ankeny. Representations of primes by quadratic forms, Amer. J. Math. 74 (1952), 913-919.
3. T. M. Apostol. Introduction to Analytic Number Theory, Springer Verlag, New York, 1986.
4. R. C. Baker. Diophantine Inequalities. LMS Monographs (New Series), vol. 1, Clarendon Press, Oxford, 1986.
5. R. C. Baker and G. Harman. On the distribution of αpk modulo one, Mathematika 38 (1991), 170-184.
6. R. C. Baker and G. Harman. Numbers with a large prime factor, Acta Arith. 73 (1995), 119-145.
7. R. C. Baker and G. Harman. The Brun-Titchmarsh Theorem on Average, Progress in Mathematics 138, Birkhäuser, Boston, 1996.
8. R. C. Baker and G. Harman. The difference between consecutive primes, Proc. London Math. Soc. (3) 72 (1996), 261-280.
9. R. C. Baker and G. Harman. Shifted primes without large prime factors, Acta Arith. 83 (1998), 331-361.
10. R. C. Baker and G. Harman. The three primes theorem with almost equal summands, Phil. Trans. R. Soc. Lond. Ser. Math.Phys. Eng. Sci. A 356 (1998), 763-780.
11. R. C. Baker, G. Harman, and J. Pintz. The exceptional set for Goldbach's problem in short intervals, Sieve Methods, Exponential Sums and Their Applications in Number Theory, Cambridge University Press, Cambridge, 1997, pp. 11-54.
12. R. C. Baker, G. Harman, and J. Pintz. The difference between consecutive primes II, Proc. London Math. Soc. (3) 83 (2001), 532-562.
13. c]. J. Number Theory 50 (1995), 261-277.
14. A. Balog. On the distribution of pθ mod 1, Acta Math. Hungar. 45 (1985), 179-199.
15. E. Bombieri. On the large sieve, Mathematika 12 (1965), 201-225.
16. E. Bombieri, J. B. Friedlander, and H. Iwaniec. Primes in arithmetic progressions to large moduli, Acta Math. 156 (1986), 203-251.
17. E. Bombieri, J. B. Friedlander, and H. Iwaniec. Primes in arithmetic progressions to large moduli II, Math. Ann. 277 (1987), 361-393.
18. E. Bombieri, J. B. Friedlander, and H. Iwaniec. Primes in arithmetic progressions to large moduli III, J. Amer. Math. Soc. 2 (1989), 215-224.
19. J. Bräudern and E. Fouvry. Le criblé vecteurs, Compos. Math. 102 (1996), 337-355.
20. V. Brun. Über das Goldbachshe Gesetz und die Anzahl der Primzahlpaare, Archiv. Math. Naturvid. B 34 (1915), 1-19.
21. J. -R. Chen. On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157-176.
22. M. D. Coleman. The distribution of points at which binary quadratic forms are prime, Proc. London Math. Soc. (3) 61 (1990), 433-456.
23. M. D. Coleman. A zero-free region for the Hecke L-functions, Mathematika 37 (1990), 287-304.
24. M. D. Coleman. The Rosser-Iwaniec sieve in number fields, with an application, Acta Arith. 65 (1993), 53-83.
25. M. D. Coleman. Relative norms of prime ideals in small regions, Mathematika 43 (1996), 40-62.
26. S. Daniel. On the divisor-sum problem for binary forms, J. Reine Angew. Math., 507 (1999), 107-129.
27. H. Davenport. Multiplicative Number Theory, 2nd ed. (revised by H. L. Montgomery), Springer, New York, 1980.
28. H. Davenport and H. Heilbronn. On indefinite quadratic forms in five variables, J. London Math. Soc. 21 (1946), 185-193.
29. C. -J. de la Vallée Poussin. Recherches analytiques sur la théorie des nombres premiers, Ann. Soc. Scient. Bruxelles 20 (1896), 183-256.
30. W. Duke, J. B. Friedlander, and H. Iwaniec. Equidistribution of roots of a quadratic congruence to prime moduli, Ann. of Math. 141 (1995), 423-441.
31. E. Fogels. Über die Ausmenullstelle der Heckeschen L-Funktion, Acta Arith. 8 (1963), 307-309.
32. K. Ford. Vinogradov's integral and bounds for the Riemann zeta function, Proc. London Math. Soc. (3) 85 (2002), 565-633.
33. K. Ford. On Bombieri's asymptotic sieve, Trans. Amer. Math. Soc. 357 (2005), 1663-1674.
34. E. Fouvry. Théme de Brun-Titchmarsh; application au théme de Fermat, Invent. Math. 79 (1985), 383-407.
35. E. Fouvry and H. Iwaniec. Exponential sums with monomials, J. Number Theory 33 (1989), 311-333.
36. E. Fouvry and H. Iwaniec. Gaussian primes, Acta Arith. 79 (1997), 249-287.
37. J. Friedlander and H. Iwaniec. The polynomial x2 +y4 captures its primes, Ann. of Math. 148 (1998), 945-1040.
38. J. Friedlander and H. Iwaniec. Asymptotic sieve for primes, Ann. Math. 148 (1998), 1041-1065.
39. J. Friedlander and H. Iwaniec. Exceptional characters and prime numbers in arithmetic progressions, Int. Math. Res. Not. 37 (2003), 2033-2050.
40. J. Friedlander and H. Iwaniec. Exceptional characters and prime numbers in short intervals, Selecta Math. (N.S.) 10 (2004), 61-69.
41. J. Friedlander and H. Iwaniec. The illusory sieve, Int. J. Number Theory (2005), 1, 459-494.
42. A. Fujii, P. X. Gallagher and H. L. Montgomery. Some hybrid bounds for character sums and Dirichlet L-series, Topics in number theory, (Proc. Colloq., Debrecen, 1974), pp. 41-57. Colloq. Math. Soc. János Bolyai, Vol. 13, North-Holland, Amsterdam 1976.
43. P. X. Gallagher. The large sieve, Mathematika 14 (1967), 14-20.
44. P. X. Gallagher. A large sieve density estimate near σ =1, Invent. Math. 11 (1970), 329-339.
45. 2 modulo 1, Proc. London Math. Soc. (3) 42 (1981), 252-269.
46. M. Goldfeld. On the number of primes p for which p +a has a large prime factor, Mathematika 16 (1969), 23-27.
47. D. A. Goldston. On Bombieri and Davenport's theorem concerning small gaps between primes, Mathematika, 39 (1992), 10-17.
48. D. A. Goldston, Y. Motohasi, J. Pintz and C. Yildirim. Small gaps between primes exist, Proc. Japan Math. Soc. 82 (2006), 61-65.
49. S. W. Graham and G. Kolesnik. Van der Corput's method of Exponential Sums, LMS LEcture Note Series 126, Cambridge University Press.
50. G. R. H. Greaves. Large prime factors of binary forms, J. Number Theory, 3 (1971), 35-59.
51. G. R. H. Greaves. Sieves in Number Theory, Springer-Verlag Berlin 2001.
52. B. J. Green and T. C. Tao. The primes contain arbitrarily long arithmetic progressions, Ann. Math. (in press). Preprint available at http://front.math.ucdavis.edu/math.NT/0404188.
53. B. J. Green and T. C. Tao. Linear equations in primes. Preprint available at http://front.math.ucdavis.edu/math.NT/0606088.
54. J. Hadamard. Sur la distribution des źde la fonction ζ(s) eteros ses conséquences arithmétiques, Bull. Soc. Math. France 24 (1896) 199-220.
55. H. Halberstam and H. -E. Richert, Sieve Methods, Academic Press, London, 1974.
56. G. H. Hardy and J. E. Littlewood, Some problems of "Partitio Numerorum" III; On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1-70.
57. G. H. Hardy and E. M. Wright. An introduction to the Theory of Numbers, 5th ed., Oxford University Press, Oxford, 1979.
58. G. Harman. Primes in short intervals, Math. Zeit. 180 (1982), 335-348.
59. G. Harman. On the distribution of αp modulo one, J. London Math. Soc. (2) 27 (1983), 9-18.
60. G. Harman. On the distribution of p modulo one, Mathematika 30 (1983), 104-116.
61. G. Harman. Numbers badly approximable by fractions with prime denominator, Math. Proc. Camb. Phil. Soc. 118 (1995), 1-5.
62. G. Harman. On the distribution of αp modulo one II, Proc. London Math. Soc. (3) 72 (1996), 241-260.
63. G. Harman. Trigonometric sums over primes III, J.Théorie Nombres Bordeaux 15 (2003), 727-740.
64. G. Harman. On ternary quadratic forms in prime variables, Mathematika 51 (2004), 83-96.
65. G. Harman. On the number of Carmichael numbers up to x, Bulletin London Math. Soc. 37 (2005), 641-650.
66. G. Harman. On the greatest prime factor of p - 1 with effective constants, Math. Comp. 74 (2005), 2035-2041.
67. G. Harman and A. Kumchev. On sums of squares of primes Math. Proc. Cambridge Phil. Soc. 140 (2006), 1-13.
68. G. Harman, A. Kumchev, and P. A. Lewis. The distribution of prime ideals of imaginary quadratic fields, Trans. Amer. Math. Soc. 356 (2003), 599-620.
69. G. Harman and P. A. Lewis. Gaussian primes in narrow sectors, Mathematika 48 (2001), 119-135.
70. G. Harman, N. Watt, and K. C. Wong. A new mean-value result for Dirichlet L-functions and polynomials, Quart. J. Math. 55 (2004), 307-324.
71. J. K. Haugland. Application of sieve methods to prime numbers, D.Phil thesis, University of Oxford, 1998.
72. D. R. Heath-Brown. The differences between consecutive primes III, J. London Math. Soc. (2) 20 (1979), 177-178.
73. D. R. Heath-Brown. Prime numbers in short intervals and a generalized Vaughan identity, Can. J. Math. 34 (1982), 1365-1377.
74. D. R. Heath-Brown. Sieve identities and gaps between primes, Astérisque 94 (1982), 61-65.
75. D. R. Heath-Brown. The Pjateckii-Sapiro prime number theorem, J. Number Theory 16 (1983), 242-266.
76. D. R. Heath-Brown. Prime twins and Siegel zeros, Proc. London Math. Soc. (3), 47 (1983), 193-224.
77. D. R. Heath-Brown. Diophantine approximation with square-free numbers, Math. Zeit. 187 (1984), 335-344.
78. D. R. Heath-Brown. The ternary Goldbach problem, Rev. Mat. Iberoamericana 1 (1985), 45-59.
79. D. R. Heath-Brown. The divisor function d3(n) in arithmetic progressions, Acta Arith. 47 (1986), 29-56.
80. D. R. Heath-Brown. The number of primes in a short interval, J. Reine Angew. Math. 389 (1988), 22-63.
81. D. R. Heath-Brown. Zero-free regions for Dirichlet L-functions and the least prime in an arithmetic progression, Proc. London Math. Soc. (3) 64 (1992), 265-338.
82. 3, Acta Math. 186 (2001), 1-84.
83. D. R. Heath-Brown. The solubility of diagonal cubic Diophantine equations, Proc. London Math. Soc. (3) 79 (1999), 241-259.
84. D. R. Heath-Brown. Lectures on Sieves, Proceedings of the Session in Analytic Number Theory and Diophantine Equations, 50 pp., Bonner Math. Schriften 360, Univ. Bonn, Bonn, 2003.
85. D. R. Heath-Brown and H. Iwaniec. On the difference between consecutive primes, Invent. Math. 55 (1979), 49-69.
86. D. R. Heath-Brown and Jia Chaohua. The largest prime factors of the integers in an interval, II, J. reine angew. Math. 498 (1998) 35-59.
87. D. R. Heath-Brown and Jia Chaohua. On the distribution of αp modulo one, Proc. London Math. Soc. (3) 84 (2002), 79-104.
88. D. R. Heath-Brown and B. Z. Moroz. Primes represented by binary cubic forms, Proc. London Math. Soc. (3) 84 (2002), 257-288.
89. D. R. Heath-Brown and B. Z. Moroz. On the representation of primes by cubic polynomials in two variables, Proc. London Math. Soc. (3) 88 (2004), 289-312.
90. E. Hecke. Eine neue Art von Zetafunctionen und ihre Beziehung zur Verteilung der Primzahlen , II, Math. Zeit. 1 (1918), 357-376; 6 (1920), 11-51.
91. A. Hildebrand. On the number of positive integers ≤ x and free of prime factors >y, J. Number Theory 22 (1986), 265-290.
92. G. Hoheisel. Primzahlprobleme in der Analysis, Sitz. Preuss. Akad. Wiss. 2 (1930), 1-13.
93. C. Hooley. Applications of sieve methods, Cambridge Tracts in Mathematics No. 70, Cambridge University Press, Cambridge, 1976.
94. C. Hooley. On a problem of Hardy and Littlewood, Acta Arith. 79 (1997), 289-311.
95. M. N. Huxley. The large sieve inequality for algebraic number fields, Mathematika, 15 (1968), 178-187.
96. M. N. Huxley. The large sieve inequality for algebraic number fields II, Proc. London Math. Soc. 21 (1970), 108-128.
97. M. N. Huxley. On the difference between consecutive primes, Invent. Math. 15 (1972), 164-170.
98. M. N. Huxley and N. Watt. Hybrid bounds for Dirichlet's L-function, Math. Proc. Cambridge Philos. Soc. 129 (2000), 385-415.
99. A. E. Ingham. On the estimation of N(σ, T ), Quart. J. Math. Oxford 8 (1940), 291-292.
100. H. Iwaniec. Primes represented by quadratic polynomials in two variables, Acta Arith. 24 (1974), 435-459.
101. H. Iwaniec. On sums of two norms from cubic fields, Journées de théorie additive des nombres, Université de Bordeaux 1 (1977), 71-89.
102. H. Iwaniec. Rosser's sieve, Acta Arith. 36 (1980), 171-202.
103. H. Iwaniec. A new form of the error term in the linear sieve, Acta Arith. 37 (1980), 307-320.
104. H. Iwaniec. Sieve Methods, Graduate Course, Rutgers university, New brunswick, NJ, unpublished notes, 1996.
105. H. Iwaniec and M. Jutila. Primes in short intervals, Ark. Mat. 17 (1979), 167-176.
106. H. Iwaniec and E. Kowalski. Analytic Number Theory, A.M.S. Colloquium Publications 53, American Mathematical Society, Providence, RI, 2004.
107. H. Iwaniec and J. Pintz. Primes in short intervals, Monatsh. Math. 98 (1984), 115-143.
108. Jia Chaohua. Goldbach numbers in a short interval (I), Sci. China 38 (1995), 385-406.
109. Jia Chaohua. Goldbach numbers in a short interval (II), Sci. China 38 (1995) 513-523.
110. Jia Chaohua. On the exceptional set of Goldbach numbers in a short interval, Acta Arith. 77 (1996), 207-287.
111. Jia Chaohua. Almost all short intervals containing prime numbers, Acta Arith. 76 (1996), 21-84.
112. Jia Chaohua and Ming-Chit Liu. On the largest prime factor of integers, Acta Arith. 95 (2000), 17-48.
113. M. Jutila. On numbers with a large prime factor, J. Indian Math. Soc (N.S.) 37 (1973), 43-53.
114. J. P. Kubilius. On a problem in the n-dimensional analytic theory of numbers, Viliniaus Valst. Univ. Mokslo dardai Fiz. Chem. Moksly Ser. 4 (1955), 5-43.
115. A. Kumchev. On the distribution of prime numbers of the form [nc], Glasg. Math. J. 41 (1999), 85-102.
116. A. Kumchev. A Diophantine inequality involving prime powers, Acta Arith. 89 (1999), 311-330.
117. A. Kumchev. The difference between consecutive primes in an arithmetic progression, Quart. J. Math. 53 (2002), 479-501.
118. A, Kumchev and M. B. S. Laporta. On a binary Diophantine inequality involving prime powers, Number Theory for the Millennium II (Urbana, IL, 2000), 307329, A. K. Peters, Natick, MA, 2002.
119. P. Lewis. Finding Gaussian primes by analytic number theory sieve methods, Ph.D. thesis, Cardiff University, 2002.
120. H. -Q. Liu and J. Wu. Numbers with a large prime factor, Acta Arith. 89 (1999) 163-187.
121. H. Li. Goldbach numbers in short intervals, Sci. China 38 (1995), 641-652.
122. Yu. V. Linnik. The Dispersion Method in Binary Additive Problems, American Mathematical Society, Providence, RI, 1963.
123. L. Shituo and Y. Qi. A Chebyshev's type of prime number theorem in a short interval II, Hardy-Ramanujan J. 15 (1992), 1-33.
124. K. Matomäki. Sums of differences between consecutive primes, Preprint, Royal Holloway, Universty of London, 2006.
125. H. Matsui. A bound for the least Gaussian prime ωwith α
126. J. Merlin. Sur quelques théorémes d'arithmétique et un énoncé qui les contient, C.R. Acad. Sci. Paris 153 (1911), 516-518.
127. F. Mertens. Ein Beitrag zur analytische Zahlentheorie, J. Reine Angew. Math. 78 (1874), 46-62.
128. H. Mikawa. On the exceptional set in Goldbach's problem, Tsukuba J. Math. 16 (1992), 513-543.
129. H. Mikawa. On primes in arithmetic progessions, Tsukuba J. Math. 25 (2001), 121-153.
130. T. Mitsui. Generalized prime number theorem, Japan J. Math. 26 (1956), 1-42.
131. H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Mathematics 227, Springer-Verlag, Berlin-New York, 1971.
132. H. L. Montgomery. The analytic principle of the large sieve, Bull. Amer. Math. Soc. 84 (1978), 547-567.
133. H.L. Montgomery. Ten lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS Number 84, American Mathematical Society, Providence, RI, 1994.
134. H. L. Montgomery and R. C. Vaughan. On the large sieve, Mathematika 20 (1973), 119-134.
135. H. L. Montgomery and R. C. Vaughan. Hilbert's Inequality, J. London Math. Soc. (2) 8 (1974), 73-82.
136. H. L. Montgomery and R. C. Vaughan. The exceptional set in Goldbach's problem, Acta Arith. 27 (1975), 353-370.
137. Y. Motohashi. A note on the least prime in an arithmetic progression with a prime difference, Acta Arith. 17 (1970), 283-285.
138. W. Narkiewicz. Elementary and Analytic Theory of Algebraic Numbers, Monografie Matematyczne 57, PWN-Polish Scientific Publishers, Warsaw, 1974.
139. A. S. Peck. Differences between consecutive primes, Proc. London Math. Soc. (3) 76 (1998), 33-69.
140. A.S. Peck. On the differences between consecutive primes, D.Phil. thesis, University of Oxford, 1996.
141. A. Perelli and J. Pintz. On the exceptional set for Goldbach's problem in short intervals, J. London Math. Soc. (2) 47 (1993), 41-49.
142. A. Perelli, J. Pintz, and S. Salerno. Bombieri's theorem in short intervals, Ann. Scuola Norm. Sup. Pisa 11 (1984), 529-538.
143. A. G. Postnikov. Introduction to Analytic Number Theory, Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1988.
144. K. Prachar. Primzahlverteilung, Springer, Berlin, 1957.
145. H. Rademacher. On the Phragmen-Lindelöf theorem and some applications, Math. Z. 72 (1959/60), 192-204.
146. K. Ramachandra. Two remarks in prime number theory, Bull. Soc. Math. France 105 (1977), 433-437.
147. K. Ramachandra. A simple proof of the mean fourth power estimate for (1/2+it)and L (1/2+it,X), Ann. Scuola Norm. Sup. Pisa (4) 1 (1974), 81-97.
148. S. Ricci. Local distribution of primes, Ph.D. Thesis, University of Michigan, Ann Arbor, 1976.
149. B. Riemann. Über die Anzhal der Primzhalen unter einer gegebenen Größe, Monatsb. Berliner Akad. (1858/60), 671-680.
150. c], Glasg. Math. J. 43 (2001), 237-254.
151. O. Robert and P. Sargos. Three-dimensional exponential sums with monomials, J. Reine Angew. Math. 591 (2006), 1-20.
152. B. Rousselet. Inés de type Brun-Titchmarsh en moyenne, Groupe de egalittravaile en théorie analytique et elementaire des nombres 1986-1987, Université de Paris-Sud (1988), pp. 91-123.
153. B. Saffari and R. C. Vaughan. On the fractional parts of x/n and related sequences II, Ann. Inst. Fourier 27 (1977), 1-30.
154. A. Selberg. On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943), 87-105.
155. E. M Stein and G. Weiss. Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton NJ, 1971.
156. I. N. Stewart and D. O. Tall. Algebraic Number Theory, Chapman and Hall/CRC, London, 2000.
157. E. C. Titchmarsh. The Theory of the Riemann Zeta-Function, 2nd ed. (revised by D. R. Heath-Brown) , Oxford University Press, Oxford, 1986.
158. R. C. Vaughan. Mean-value theorems in prime number theory, J. London Math. Soc. (2) 10 (1975), 53-162.
159. R. C. Vaughan. An elementary method in prime number theory, Acta Arith. 37 (1980), 111-115.
160. R. C. Vaughan. An elementary method in prime number theory., Recent Progress in Analytic Number Theory, vol. 1 (Durham, 1979), Academic Press, London-New York, 1981, pp. 341-348.
161. R. C. Vaughan. The Hardy-Littlewood method, Cambridge Tracts in Mathematics 80, Cambridge University Press, Cambridge, 1981.
162. A. I. Vinogradov. On the density hypothesis for Dirichlet L-functions, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 903-934.
163. I. M. Vinogradov. The method of trigonometric sums in the theory of numbers. Translated from the Russian, revised, and annotated by K. F. Roth and A. Davenport. Reprint of the 1954 translation. Dover, Mineola, NY, 2004.
164. N. Watt. Kloosterman sums and a mean value for Dirichlet polynomials, J. Number Theory 53 (1995), 179-210.
165. N. Watt. Short intervals almost all containing primes, Acta Arith. 72 (1995), 131-167.
166. D. Wolke. Grosse Differenzen aufeinanderfolgender primzahlen, Math. Ann. 218 (1975), 269-271.
167. K. C.Wong. Contributions to analytic number theory, Ph.D. thesis, Cardiff University, 1996.
168. J. Wu. On the primitive circle problem, Monatsch. Math. 135 (2002), 69-81.