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A variant of the Bombieri-Vinogradov theorem with explicit constants and applications. (English) Zbl 1341.11053

The authors give an effective version with explicit constants of a mean value theorem of R. C. Vaughan [J. Lond. Math. Soc., II. Ser. 6, 43–55 (1972; Zbl 0254.10040); ibid. II. Ser. 6, 326–328 (1973; Zbl 0254.10041)] related to the values of the twisted summatory function associated to the von Mangoldt function and a Dirichlet character. As a consequence, they prove an effective variant of the Bombieri-Vinogradov theorem with explicit constants. This effective variant can be used to obtain explicit results in many problems: as examples, the authors provide such results in several number theoretical problems related to shifted primes.

MSC:

11N13 Primes in congruence classes
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References:

[1] Bombieri, E., On the large sieve, Mathematika, 12, 201-225 (1965) · Zbl 0136.33004
[2] Baker, R. C.; Harman, G., Shifted primes without large prime factors, Acta Arith., 83, 4, 331-361 (1998) · Zbl 0994.11033
[3] Cojocaru, Alina Carmen; Murty, M. Ram, An Introduction to Sieve Methods and Their Applications, London Mathematical Society Student Texts 66, xii+224 pp. (2006), Cambridge University Press: Cambridge:Cambridge University Press · Zbl 1121.11063
[4] Davenport, Harold, Multiplicative Number Theory, Graduate Texts in Mathematics 74, xiv+177 pp. (2000), Springer-Verlag: New York:Springer-Verlag · Zbl 1002.11001
[5] Dress, F.; Iwaniec, H.; Tenenbaum, G., Sur une somme li\'ee \`“a la fonction de M\'”obius, J. Reine Angew. Math., 340, 53-58 (1983) · Zbl 0497.10003
[6] Dusart, Pierre, In\'egalit\'es explicites pour \(\psi (X), \theta (X), \pi (X)\) et les nombres premiers, C. R. Math. Acad. Sci. Soc. R. Can., 21, 2, 53-59 (1999) · Zbl 0935.11002
[7] [E] P. Erd\`“os, On the normal number of prime factors of \(p-1\) and some related problems concerning Euler”s \(\phi \)-function, Quart. J. Math. 6 (1935), 205-213.
[8] Goldfeld, Morris, On the number of primes \(p\) for which \(p+a\) has a large prime factor, Mathematika, 16, 23-27 (1969) · Zbl 0201.05301
[9] Granville, Andrew; Ramar{\'e}, Olivier, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika, 43, 1, 73-107 (1996) · Zbl 0868.11009
[10] Harman, G., On the greatest prime factor of \(p-1\) with effective constants, Math. Comp., 74, 252, 2035-2041 (electronic) (2005) · Zbl 1101.11037
[11] [LP] H. W. Lenstra and C. Pomerance, Primality testing with Gaussian periods, preprint, 47 pages. · Zbl 1429.11221
[12] Montgomery, Hugh L., The analytic principle of the large sieve, Bull. Amer. Math. Soc., 84, 4, 547-567 (1978) · Zbl 0408.10033
[13] Montgomery, H. L.; Vaughan, R. C., The large sieve, Mathematika, 20, 119-134 (1973) · Zbl 0296.10023
[14] Motohashi, Yoichi, A note on the least prime in an arithmetic progression with a prime difference, Acta Arith., 17, 283-285 (1970) · Zbl 0215.35302
[15] Murty, M. Ram, Problems in Analytic Number Theory, Graduate Texts in Mathematics 206, xvi+452 pp. (2001), Springer-Verlag: New York:Springer-Verlag · Zbl 0971.11001
[16] Pomerance, Carl, Remarks on the P\'olya-Vinogradov inequality, Integers, 11, 4, 531-542 (2011) · Zbl 1278.11080
[17] Rosser, J. Barkley; Schoenfeld, Lowell, Approximate formulas for some functions of prime numbers, Illinois J. Math., 6, 64-94 (1962) · Zbl 0122.05001
[18] Timofeev, N. M., The Vinogradov-Bombieri theorem, Mat. Zametki, 38, 6, 801-809, 956 (1985) · Zbl 0588.10043
[19] Vaughan, R. C., On the number of solutions of the equation \(p=a+n_1\cdots n_k\) with \(a<p\leq x\), J. London Math. Soc. (2), 6, 43-55 (1972) · Zbl 0254.10040
[20] Vaughan, R. C., On the number of solutions of the equation \(p+n_1\cdots n_k=N\), J. London Math. Soc. (2), 6, 326-328 (1973) · Zbl 0254.10041
[21] Vaughan, R. C., Sommes trigonom\'etriques sur les nombres premiers, C. R. Acad. Sci. Paris S\'er. A-B, 285, 16, A981-A983 (1977)
[22] Vaughan, R. C., An elementary method in prime number theory, Acta Arith., 37, 111-115 (1980) · Zbl 0448.10037
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