A variant of the Bombieri-Vinogradov theorem with explicit constants and applications
Authors:
Amir Akbary and Kyle Hambrook
Journal:
Math. Comp. 84 (2015), 1901-1932
MSC (2010):
Primary 11N13
DOI:
https://doi.org/10.1090/S0025-5718-2014-02919-0
Published electronically:
December 29, 2014
MathSciNet review:
3335897
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We give an effective version with explicit constants of a mean value theorem of Vaughan related to the values of $\psi (y, \chi )$, the twisted summatory function associated to the von Mangoldt function $\Lambda$ and a Dirichlet character $\chi$. As a consequence of this result we prove an effective variant of the Bombieri-Vinogradov theorem with explicit constants. This effective variant has the potential to provide explicit results in many problems. We give examples of such results in several number theoretical problems related to shifted primes.
- E. Bombieri, On the large sieve, Mathematika 12 (1965), 201–225. MR 197425, DOI https://doi.org/10.1112/S0025579300005313
- R. C. Baker and G. Harman, Shifted primes without large prime factors, Acta Arith. 83 (1998), no. 4, 331–361. MR 1610553, DOI https://doi.org/10.4064/aa-83-4-331-361
- Alina Carmen Cojocaru and M. Ram Murty, An introduction to sieve methods and their applications, London Mathematical Society Student Texts, vol. 66, Cambridge University Press, Cambridge, 2006. MR 2200366
- 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
- F. Dress, H. Iwaniec, and G. Tenenbaum, Sur une somme liée à la fonction de Möbius, J. Reine Angew. Math. 340 (1983), 53–58 (French). MR 691960, DOI https://doi.org/10.1515/crll.1983.340.53
- Pierre Dusart, Inégalités explicites pour $\psi (X)$, $\theta (X)$, $\pi (X)$ et les nombres premiers, C. R. Math. Acad. Sci. Soc. R. Can. 21 (1999), no. 2, 53–59 (French, with English and French summaries). MR 1697455
- P. Erdös, 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.
- Morris Goldfeld, On the number of primes $p$ for which $p+a$ has a large prime factor, Mathematika 16 (1969), 23–27. MR 244176, DOI https://doi.org/10.1112/S0025579300004575
- Andrew Granville and Olivier Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), no. 1, 73–107. MR 1401709, DOI https://doi.org/10.1112/S0025579300011608
- G. Harman, On the greatest prime factor of $p-1$ with effective constants, Math. Comp. 74 (2005), no. 252, 2035–2041. MR 2164111, DOI https://doi.org/10.1090/S0025-5718-05-01749-7
- H. W. Lenstra and C. Pomerance, Primality testing with Gaussian periods, preprint, 47 pages.
- Hugh L. Montgomery, The analytic principle of the large sieve, Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. MR 466048, DOI https://doi.org/10.1090/S0002-9904-1978-14497-8
- H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119–134. MR 374060, DOI https://doi.org/10.1112/S0025579300004708
- Yoichi Motohashi, A note on the least prime in an arithmetic progression with a prime difference, Acta Arith. 17 (1970), 283–285. MR 268131, DOI https://doi.org/10.4064/aa-17-3-283-285
- M. Ram Murty, Problems in analytic number theory, Graduate Texts in Mathematics, vol. 206, Springer-Verlag, New York, 2001. Readings in Mathematics. MR 1803093
- Carl Pomerance, Remarks on the Pólya-Vinogradov inequality, Integers 11 (2011), no. 4, 531–542. MR 2988079, DOI https://doi.org/10.1515/integ.2011.039
- J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
- N. M. Timofeev, The Vinogradov-Bombieri theorem, Mat. Zametki 38 (1985), no. 6, 801–809, 956 (Russian). MR 823418
- R. C. Vaughan, 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 (1972), 43–55. MR 309889, DOI https://doi.org/10.1112/jlms/s2-6.1.43
- R. C. Vaughan, On the number of solutions of the equation $p+n_{1}\cdots n_{k}=N$, J. London Math. Soc. (2) 6 (1973), 326–328. MR 309890, DOI https://doi.org/10.1112/jlms/s2-6.2.326
- Robert-C. Vaughan, Sommes trigonométriques sur les nombres premiers, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 16, A981–A983 (French, with English summary). MR 498434
- R. C. Vaughan, An elementary method in prime number theory, Acta Arith. 37 (1980), 111–115. MR 598869, DOI https://doi.org/10.4064/aa-37-1-111-115
Retrieve articles in Mathematics of Computation with MSC (2010): 11N13
Retrieve articles in all journals with MSC (2010): 11N13
Additional Information
Amir Akbary
Affiliation:
Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB T1K 3M4 Canada
MR Author ID:
650700
Email:
amir.akbary@uleth.ca
Kyle Hambrook
Affiliation:
Department of Mathematics, 1984 Mathematics Road, University of British Columbia, Vancouver, BC V6T 1Z2 Canada
MR Author ID:
952267
ORCID:
0000-0002-0097-4257
Email:
hambrook@math.ubc.ca
Keywords:
Bombieri-Vinogradov theorem,
divisors of shifted primes
Received by editor(s):
February 27, 2013
Received by editor(s) in revised form:
November 5, 2013
Published electronically:
December 29, 2014
Additional Notes:
The research of the authors was partially supported by NSERC and ULRF
Article copyright:
© Copyright 2014
American Mathematical Society