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Proceedings of the Steklov Institute of Mathematics

, Volume 299, Issue 1, pp 219–245 | Cite as

Sums of Values of Nonprincipal Characters over a Sequence of Shifted Primes

  • Z. Kh. Rakhmonov
Article
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Abstract

For a nonprincipal character χ modulo D, we prove a nontrivial estimate of the form Σnx Λ(n)χ(n − l) \( \ll x\exp \{ - 0.6\sqrt {\ln D} \} \) for the sum of values of χ over a sequence of shifted primes in the case when xD1/2+ε, (l,D) = 1, and the modulus of the primitive character generated by χ is a cube-free number.

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References

  1. 1.
    D. A. Burgess, “On character sums and L-series,” Proc. London Math. Soc., Ser. 3, 12, 193–206 (1962).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    D. A. Burgess, “On character sums and L-series. II,” Proc. London Math. Soc., Ser. 3, 13, 524–536 (1963).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    J. B. Friedlander, K. Gong, and I. E. Shparlinskii, “Character sums over shifted primes,” Mat. Zametki 88 (4), 605–619 (2010) [Math. Notes 88, 585–598 (2010)].MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    A. A. Karatsuba, “Character sums and primitive roots in finite fields,” Dokl. Akad. Nauk SSSR 180 (6), 1287–1289 (1968) [Sov. Math., Dokl. 9, 755–757 (1968)].MathSciNetMATHGoogle Scholar
  5. 5.
    A. A. Karatsuba, “Estimates of character sums,” Izv. Akad. Nauk SSSR, Ser. Mat. 34 (1), 20–30 (1970) [Math. USSR, Izv. 4 (1), 19–29 (1970)].MathSciNetGoogle Scholar
  6. 6.
    A. A. Karatsuba, “Sums of characters over prime numbers,” Izv. Akad. Nauk SSSR, Ser. Mat. 34 (2), 299–321 (1970) [Math. USSR, Izv. 4 (2), 303–326 (1970)].MathSciNetMATHGoogle Scholar
  7. 7.
    A. A. Karatsuba, “On sums of characters with primes,” Dokl. Akad. Nauk SSSR 190 (3), 517–518 (1970) [Sov. Math., Dokl. 11, 135–137 (1970)].MathSciNetMATHGoogle Scholar
  8. 8.
    A. A. Karatsuba, “The distribution of products of shifted primes in arithmetical progressions,” Dokl. Akad. Nauk SSSR 192 (4), 724–727 (1970) [Sov. Math., Dokl. 11, 707–711 (1970)].MathSciNetMATHGoogle Scholar
  9. 9.
    A. A. Karatsuba, “Sums of characters with prime numbers in an arithmetic progression,” Izv. Akad. Nauk SSSR, Ser. Mat. 35 (3), 469–484 (1971) [Math. USSR, Izv. 5 (3), 485–501 (1971)].MathSciNetMATHGoogle Scholar
  10. 10.
    A. A. Karatsuba, “Sums of characters in sequences of shifted prime numbers, with applications,” Mat. Zametki 17 (1), 155–159 (1975) [Math. Notes 17, 91–93 (1975)].MathSciNetMATHGoogle Scholar
  11. 11.
    A. A. Karatsuba, “Some problems of contemporary analytic number theory,” Mat. Zametki 17 (2), 341–350 (1975) [Math. Notes 17, 195–199 (1975)].MathSciNetMATHGoogle Scholar
  12. 12.
    A. A. Karatsuba, “On the distribution of values of nonprincipal characters,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 142, 156–164 (1976) [Proc. Steklov Inst. Math. 142, 165–174 (1979)].MathSciNetMATHGoogle Scholar
  13. 13.
    A. A. Karatsuba, “Sums of Legendre symbols of polynomials of second degree over prime numbers,” Izv. Akad. Nauk SSSR, Ser. Mat. 42 (2), 315–324 (1978) [Math. USSR, Izv. 12 (2), 299–308 (1978)].MathSciNetMATHGoogle Scholar
  14. 14.
    A. A. Karatsuba, Basic Analytic Number Theory (Nauka, Moscow, 1983; Springer, Berlin, 1993).Google Scholar
  15. 15.
    A. A. Karatsuba, “Arithmetic problems in the theory of Dirichlet characters,” Usp. Mat. Nauk 63 (4), 43–92 (2008) [Russ. Math. Surv. 63, 641–690 (2008)].MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Yu. V. Linnik, “Recent works of I. M. Vinogradov,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 132, 27–29 (1973) [Proc. Steklov Inst. Math. 132, 25–28 (1973)].MathSciNetGoogle Scholar
  17. 17.
    Z. Kh. Rakhmonov, “On the distribution of values of Dirichlet characters,” Usp. Mat. Nauk 41 (1), 201–202 (1986) [Russ. Math. Surv. 41 (1), 237–238 (1986)].MathSciNetMATHGoogle Scholar
  18. 18.
    Z. Kh. Rakhmonov, “On an estimate for a character sum over primes,” Dokl. Akad. Nauk Tadzhik. SSR 29 (1), 16–20 (1986).MathSciNetMATHGoogle Scholar
  19. 19.
    Z. Kh. Rakhmonov, “On the least Goldbach number in an arithmetic progression,” Izv. Akad. Nauk Tadzhik. SSR, Otd. Fiz.-Mat. Khim. Geol. Nauk, No. 2, 103–106 (1986).MathSciNetMATHGoogle Scholar
  20. 20.
    Z. Kh. Rakhmonov, “Theorem on the mean value of ψ(x, χ) and its applications,” Izv. Ross. Akad. Nauk, Ser. Mat. 57 (4), 55–71 (1993) [Russ. Acad. Sci., Izv. Math. 43 (1), 49–64 (1994)].MathSciNetGoogle Scholar
  21. 21.
    Z. Kh. Rakhmonov, “On the distribution of values of Dirichlet characters and their applications,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 207, 286–296 (1994) [Proc. Steklov Inst. Math. 207, 263–272 (1995)].MathSciNetMATHGoogle Scholar
  22. 22.
    Z. Kh. Rakhmonov, “Distribution of values of Dirichlet characters in the sequence of shifted primes,” Dokl. Akad. Nauk Resp. Tadzhikistan 56 (1), 5–9 (2013).MATHGoogle Scholar
  23. 23.
    Z. Kh. Rakhmonov, “Distribution of values of Dirichlet characters in a sequence of shifted primes,” Izv. Saratov. Univ., Nov. Ser., Mat. Mekh. Inf. 13 (4), part 2, 113–117 (2013).MATHGoogle Scholar
  24. 24.
    Z. Kh. Rakhmonov, “Sums of characters over prime numbers,” Chebyshev. Sb. 15 (2), 73–100 (2014).MATHGoogle Scholar
  25. 25.
    Z. Kh. Rakhmonov and Sh. Kh. Mirzorakhimov, “Sums of characters modulo a cubefree at shifted primes,” Chebyshev. Sb. 17 (1), 201–216 (2016).Google Scholar
  26. 26.
    A. I. Vinogradov, “On numbers with small prime divisors,” Dokl. Akad. Nauk SSSR 109 (4), 683–686 (1956).MathSciNetMATHGoogle Scholar
  27. 27.
    I. M. Vinogradov, “On the distribution of quadratic rests and non-rests of the form p + k to a prime modulus,” Mat. Sb. 3 (2), 311–319 (1938).MATHGoogle Scholar
  28. 28.
    I. M. Vinogradov, “An improvement of the estimation of sums with primes,” Izv. Akad. Nauk SSSR, Ser. Mat. 7 (1), 17–34 (1943).MathSciNetGoogle Scholar
  29. 29.
    I. M. Vinogradov, “New approach to the estimation of a sum of values of χ(p + k),” Izv. Akad. Nauk SSSR, Ser. Mat. 16 (3), 197–210 (1952).MathSciNetMATHGoogle Scholar
  30. 30.
    I. M. Vinogradov, “Improvement of an estimate for the sum of the values χ(p + k),” Izv. Akad. Nauk SSSR, Ser. Mat. 17 (4), 285–290 (1953).MathSciNetGoogle Scholar
  31. 31.
    I. M. Vinogradov, “An estimate for a certain sum extended over the primes of an arithmetic progression,” Izv. Akad. Nauk SSSR, Ser. Mat. 30 (3), 481–496 (1966).MathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.A. Juraev Institute of MathematicsAcademy of Sciences of the Republic of TajikistanDushanbeTajikistan

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