On the Brun-Titchmarsh theorem
Henryk IWANIEC
著者情報
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Henryk IWANIEC
Université de Bordeaux I Instytut Matematyczny Polskiej Akademii Nauk
ジャーナル
フリー
1982 年 34 巻 1 号 p. 95-123
詳細
- Published: 1982 Received: 1980/05/23 Available on J-STAGE: 2006/10/20 Accepted: - Advance online publication: - Revised: -
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訂正情報
訂正日: 2006/10/20 訂正理由: - 訂正箇所: 引用文献情報 訂正内容: Wrong : 1) M.B. Barban, Yu. V. Linnik and N.G. Tshudakov, On prime numbers in an arithmetic progression with a prime-power difference, Acta Arith., 9 (1964), 375-390.
2) E. Bombieri and H. Davenport, On the large sieve method, Abh. aus Zahlentheorie und Analysis zur Erinnerung an Edmund Landau, Deutscher Verlag Wiss., Berlin 1968, 11-22.
3) D. Burgess, On character sums and L-series, Proc. London Math. Soc., (3) 12 (1962), 193-206.
4) D. Burgess, On character sums and L-series, II, Proc. London Math. Soc., (3) 13 (1963), 524-536.
5) A. Fujii, P.X. Gallagher and H.L. Montgomery, Some hybrid bounds for character sums and Dirichlet's L-series, Topics in number theory (Proc. Colloq. Debrecen (1974), 41-47), North-Holland, Amsterdam, 1976.
6) P.X. Gallagher, Primes in progressions to prime power modulus, Invent. Math., 16 (1972), 191-201.
7) D. Goldfeld, On the number of primes p for which p+a has a large prime factor, Mathematika, 16 (1969), 23-27.
8) D. Goldfeld, A further improvement of the Brun-Titchmarsh theorem, J. London Math. Soc., (2) 11 (1977), 434-444.
9) G. Greaves, A weighted sieve of Brun's type, Acta Atith., (to appears).
10) R. Heath-Brown, Almost-primes in arithmetic progressions and short intervals, Math. Proc. Cambridge Philos. Soc., 83 (1978), 357-375.
11) R. Heath-Brown and H. Iwaniec, On the difference between consecutive primes, Invent. Math., 55 (1979), 49-69.
12) C. Hooley, On the Brun-Titchmarsh theorem, J. Reine Angew. Math., 255 (1972), 60-79.
13) C. Hooley, On the largest prime factor of p+a, Mathematika, 20 (1973), 135-143.
14) C. Hooley, On the Brun Titchmarsh theorem, II, Proc. London Math. Soc., (3) 30 (1975), 114-128.
15) C. Hooley, On the greatest prime factor of a cubic polynomial, J. Reine Angew. Math., 303/304 (1978), 21-50.
16) H. Iwaniec, On the error term in the linear sieve, Acta Arith., 19 (1971), 1-30.
17) H. Iwaniec, On zeros of Dirichlet's L-series, Invent. Math., 23 (1974), 97-104.
18) H. Iwaniec, A new form of the error term in the linear sieve, Acta Arith., 37 (1980), 307-320.
19) M. Jutila, Zero density estimates for L-functions, Acta Arith., 32 (1977), 55-62.
20) M. Laborde, Nombres presque-premiers dans de petits intervalles, Séminaire de Théorie des Nombres, Bordeaux, 1977-1978.
21) M. Laborde, Buchstab's sifting weights, Mathematika, 26 (1979), 250-257.
22) J.H. van Lint and H.E. Richert, On primes in arithmetic progressions, Acta Arith., 11 (1965), 209-216.
23) H.L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math., 227, Berlin-New York, 1971.
24) H.L. Montgomery, Problems concerning prime numbers, Proc. Symp. Pure Math., 28 (1976), 307-310.
25) H.L. Montgomery and R.C. Vaughan, On the large sieve, Mathematika, 20 (1973), 119-134.
26) Y. Motohashi, A note on the least prime in an arithmetic progression with a prime difference, Acta Arith., 17 (1970), 283-285.
27) Y. Motohashi, On some improvements of the Brun-Titchmarsh theorem, J. Math. Soc. Japan, 26 (1974), 306-323.
28) Y. Motohashi, On some improvements of the Brun-Titchmarsh theorem III, J. Math. Soc. Japan, 27 (1975), 444-453.
29) Y. Motohashi, A note on Siegel's zero, Proc. Japan Acad., 55 (1979), 190-192.
30) E. Philips, The zeta-function of Riemann, Quart. J. of Math. Oxford Ser., 4 (1933), 209-225.
31) K. Prachar, Primzahlverteilung, Berlin-Gottingen-Heidelberg, 1957.
32) H.E. Richert, Selberg's sieve with weights, Mathematika, 16 (1969), 1-22.
33) H. Siebert, On a question of P. Turan, Acta Arith., 26 (1975), 303-305.
34) E.C. Titchmarsh, A divisor problem, Rend. Circ. Mat. Palermo, 54 (1930), 414-429.
35) P. Turan, Uber die Siegel-Nullestelle der Dirichletschen Funktionen, Acta Arith., 24 (1973), 135-141.
36) D, Wolke, Eine weitere Möglichkeit zur Verbesserung des Satzes von Brun-Titchmarsh (unpublished manuscript).
Right : [1] M. B. Barban, Yu. V. Linnik and N. G. Tshudakov, On prime numbers in an arithmetic progression with a prime-power difference, Acta Arith., 9 (1964), 375-390.
[2] E. Bombieri and H. Davenport, On the large sieve method, Abh. aus Zahlentheorie und Analysis zur Erinnerung an Edmund Landau, Deutscher Verlag Wiss., Berlin 1968, 11-22.
[3] D. Burgess, On character sums and L-series, Proc. London Math. Soc., (3) 12 (1962), 193-206.
[4] D. Burgess, On character sums and L-series, II, Proc. London Math. Soc., (3) 13 (1963), 524-536.
[5] A. Fujii, P. X. Gallagher and H. L. Montgomery, Some hybrid bounds for character sums and Dirichlet's L-series, Topics in number theory (Proc. Colloq. Debrecen (1974), 41-47), North-Holland, Amsterdam, 1976.
[6] P. X. Gallagher, Primes in progressions to prime power modulus, Invent. Math., 16 (1972), 191-201.
[7] D. Goldfeld, On the number of primes p for which p+a has a large prime factor, Mathematika, 16 (1969), 23-27.
[8] D. Goldfeld, A further improvement of the Brun-Titchmarsh theorem, J. London Math. Soc., (2) 11 (1977), 434-444.
[9] G. Greaves, A weighted sieve of Brun's type, Acta Atith., (to appears).
[10] R. Heath-Brown, Almost-primes in arithmetic progressions and short intervals, Math. Proc. Cambridge Philos. Soc., 83 (1978), 357-375.
[11] R. Heath-Brown and H. Iwaniec, On the difference between consecutive primes, Invent. Math., 55 (1979), 49-69.
[12] C. Hooley, On the Brun-Titchmarsh theorem, J. Reine Angew. Math., 255 (1972), 60-79.
[13] C. Hooley, On the largest prime factor of p+a, Mathematika, 20 (1973), 135-143.
[14] C. Hooley, On the Brun Titchmarsh theorem, II, Proc. London Math. Soc., (3) 30 (1975), 114-128.
[15] C. Hooley, On the greatest prime factor of a cubic polynomial, J. Reine Angew. Math., 303/304 (1978), 21-50.
[16] H. Iwaniec, On the error term in the linear sieve, Acta Arith., 19 (1971), 1-30.
[17] H. Iwaniec, On zeros of Dirichlet's L-series, Invent. Math., 23 (1974), 97-104.
[18] H. Iwaniec, A new form of the error term in the linear sieve, Acta Arith., 37 (1980), 307-320.
[19] M. Jutila, Zero density estimates for L-functions, Acta Arith., 32 (1977), 55-62.
[20] M. Laborde, Nombres presque-premiers dans de petits intervalles, Séminaire de Théorie des Nombres, Bordeaux, 1977-1978.
[21] M. Laborde, Buchstab's sifting weights, Mathematika, 26 (1979), 250-257.
[22] J. H. van Lint and H. E. Richert, On primes in arithmetic progressions, Acta Arith., 11 (1965), 209-216.
[23] H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math., 227, Berlin-New York, 1971.
[24] H. L. Montgomery, Problems concerning prime numbers, Proc. Symp. Pure Math., 28 (1976), 307-310.
[25] H. L. Montgomery and R. C. Vaughan, On the large sieve, Mathematika, 20 (1973), 119-134.
[26] Y. Motohashi, A note on the least prime in an arithmetic progression with a prime difference, Acta Arith., 17 (1970), 283-285.
[27] Y. Motohashi, On some improvements of the Brun-Titchmarsh theorem, J. Math. Soc. Japan, 26 (1974), 306-323.
[28] Y. Motohashi, On some improvements of the Brun-Titchmarsh theorem III, J. Math. Soc. Japan, 27 (1975), 444-453.
[29] Y. Motohashi, A note on Siegel's zero, Proc. Japan Acad., 55 (1979), 190-192.
[30] E. Philips, The zeta-function of Riemann, Quart. J. of Math. Oxford Ser., 4 (1933), 209-225.
[31] K. Prachar, Primzahlverteilung, Berlin-Göttingen-Heidelberg, 1957.
[32] H. E. Richert, Selberg's sieve with weights, Mathematika, 16 (1969), 1-22.
[33] H. Siebert, On a question of P. Turan, Acta Arith., 26 (1975), 303-305.
[34] E. C. Titchmarsh, A divisor problem, Rend. Circ. Mat. Palermo, 54 (1930), 414-429.
[35] P. Turan, Uber die Siegel-Nullestelle der Dirichletschen Funktionen, Acta Arith., 24 (1973), 135-141.
[36] D. Wolke, Eine weitere Möglichkeit zur Verbesserung des Satzes von Brun-Titchmarsh (unpublished manuscript).
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