Primes in Arithmetic Progressions to Moduli with a Large Power Factor

ABSTRACT

Recently Elliott studied the distribution
of primes in arithmetic progressions whose moduli can be divisible by highpowers of a given integer and showed that for
integer *a*≥2 and real number *A*＞0. There is a *B*=*B*(*A*)＞0 such that

,

holds uniformly for moduli that are powers of* a*. In this paper we are able to
improve his result.

Cite this paper

R. Guo, "Primes in Arithmetic Progressions to Moduli with a Large Power Factor,"*Advances in Pure Mathematics*, Vol. 3 No. 7, 2013, pp. 25-32. doi: 10.4236/apm.2013.37A003.

R. Guo, "Primes in Arithmetic Progressions to Moduli with a Large Power Factor,"

References

[1] H. Iwaniec and E. Kowalski, “Analytic Number Theory,” American Mathematical Society, Providence, 2004.

[2] E. Bombieri, “On the Large Sieve,” Mathematika, Vol. 12, No. 2, 1965, pp. 210-225.

http://dx.doi.org/10.1112/S0025579300005313

[3] A. I. Vinogradov, “The Density Hypothesis for Dirichlet L-Series,” Izvestiya Rossiiskoi Akademii Nauk SSSR. Seriya Matematicheskaya, Vol. 29, 1965, pp. 903-934.

http://dx.doi.org/10.1007/s11139-006-0250-4

[4] P. D. T. A. Elliott, “Primes in Progressions to Moduli with a Large Power Factor,” The Ramanujan Journal, Vol. 13, No. 1-3, 2007, pp. 241-251.

[5] M. B. Barban, Y. V. Linink and N. G. Chudakov, “On Prime Numbers in an Arithmetic Progression with a Prime-Power Difference,” Acta Arithmetica, Vol. 9, No. 4, 1964, pp. 375-390.

[6] J. Y. Liu and M. C. Liu, “The Exceptional Set in Four Prime Squares Problem,” Illinois Journal of Mathematics, Vol. 44, No. 2, 2000, pp. 272-293.

[7] J. Y. Liu, “On Lagrange’s Theorem with Prime Variables,” Quarterly Journal of Mathematics (Oxford), Vol. 54, No. 4, 2003, pp. 453-462.

http://dx.doi.org/10.1093/qmath/hag028

[8] D. R. Heath-Brown, “Prime Numbers in Short Intervals and a Generalized Vaughan’s Identity,” Canadian Journal of Mathematics, Vol. 34, 1982, pp. 1365-1377.

http://dx.doi.org/10.4153/CJM-1982-095-9

[9] S. K. K. Choi and A. Kumchev, “Mean Values of Dirichlet Polynomials and Applications to Linear Equations with Prime Variables,” Acta Arithmetica, Vol. 123, No. 2, 2006, pp. 125-142.

http://dx.doi.org/10.4064/aa123-2-2

[10] H. Ivaniec, “On Zeros of Dirichlet’s L-Series,” Inventiones Mathematicae, Vol. 23, No. 2, 1974, pp. 97-104.

http://dx.doi.org/10.1007/BF01405163

[11] K. Prachar, “Primzahlverteilung,” Grund. der math. Wiss., Springer-Verlag, Berlin-Gottingen-Heidelberg, 1957.

[12] D. R. Heath-Brown, “Zero-Free Regions for Dirichlet L-Functions and the Least Prime in an Arithmetic Progressions,” Proceedings of the London Mathematical Society, Vol. 64, No. 2, 1992, pp. 265-338.

http://dx.doi.org/10.1112/plms/s3-64.2.265

[1] H. Iwaniec and E. Kowalski, “Analytic Number Theory,” American Mathematical Society, Providence, 2004.

[2] E. Bombieri, “On the Large Sieve,” Mathematika, Vol. 12, No. 2, 1965, pp. 210-225.

http://dx.doi.org/10.1112/S0025579300005313

[3] A. I. Vinogradov, “The Density Hypothesis for Dirichlet L-Series,” Izvestiya Rossiiskoi Akademii Nauk SSSR. Seriya Matematicheskaya, Vol. 29, 1965, pp. 903-934.

http://dx.doi.org/10.1007/s11139-006-0250-4

[4] P. D. T. A. Elliott, “Primes in Progressions to Moduli with a Large Power Factor,” The Ramanujan Journal, Vol. 13, No. 1-3, 2007, pp. 241-251.

[5] M. B. Barban, Y. V. Linink and N. G. Chudakov, “On Prime Numbers in an Arithmetic Progression with a Prime-Power Difference,” Acta Arithmetica, Vol. 9, No. 4, 1964, pp. 375-390.

[6] J. Y. Liu and M. C. Liu, “The Exceptional Set in Four Prime Squares Problem,” Illinois Journal of Mathematics, Vol. 44, No. 2, 2000, pp. 272-293.

[7] J. Y. Liu, “On Lagrange’s Theorem with Prime Variables,” Quarterly Journal of Mathematics (Oxford), Vol. 54, No. 4, 2003, pp. 453-462.

http://dx.doi.org/10.1093/qmath/hag028

[8] D. R. Heath-Brown, “Prime Numbers in Short Intervals and a Generalized Vaughan’s Identity,” Canadian Journal of Mathematics, Vol. 34, 1982, pp. 1365-1377.

http://dx.doi.org/10.4153/CJM-1982-095-9

[9] S. K. K. Choi and A. Kumchev, “Mean Values of Dirichlet Polynomials and Applications to Linear Equations with Prime Variables,” Acta Arithmetica, Vol. 123, No. 2, 2006, pp. 125-142.

http://dx.doi.org/10.4064/aa123-2-2

[10] H. Ivaniec, “On Zeros of Dirichlet’s L-Series,” Inventiones Mathematicae, Vol. 23, No. 2, 1974, pp. 97-104.

http://dx.doi.org/10.1007/BF01405163

[11] K. Prachar, “Primzahlverteilung,” Grund. der math. Wiss., Springer-Verlag, Berlin-Gottingen-Heidelberg, 1957.

[12] D. R. Heath-Brown, “Zero-Free Regions for Dirichlet L-Functions and the Least Prime in an Arithmetic Progressions,” Proceedings of the London Mathematical Society, Vol. 64, No. 2, 1992, pp. 265-338.

http://dx.doi.org/10.1112/plms/s3-64.2.265