Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Rakhmonov Z. K. Distribution of Values of Dirichlet Characters in the Sequence of Shifted Primes. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2013, vol. 13, iss. 4, pp. 113-117. DOI: 10.18500/1816-9791-2013-13-4-113-117

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
25.11.2013
Full text:
download
(downloads: 145)
Language: 
Russian
Heading: 
Mathematics
UDC: 
511.325
DOI: 
10.18500/1816-9791-2013-13-4-113-117

Distribution of Values of Dirichlet Characters in the Sequence of Shifted Primes

Autors: 
Rakhmonov Zarullo Khusenovich, Dzhuraev Institute of Mathematics, Academy of Sciences of Republic of Tajikistan, Dushanbe, Tajikistan
Abstract: 

The new estimate for the sum of the values of a primitive Dirichlet character modulo an integer q has been obtained over the sequence of shifted primes p − l, (l, q) = 1, p ≤ x. This estimate is nontrivial for x ≥ q 5 6 +ε and refines the estimate obtained by J. B. Friedlander, K. Gong, I. E. Shparlinskii. Their estimate holds provided that x ≥ q 8/9+ε

Key words: 
dirichlet character
shifted primes
short sums of characters
exponential sums over primes
References: 
  1. Vinogradov I. M. On the distribution of quadratic rests and non-rests of the form p + k to a prime modulus. Rec. Math. Moscow, n. Ser., 1938, vol. 3, no. 45, pp. 311–319  (in Russian).
  2. Vinogradow I. M. An improvement of the estimation of sums with primes. Bull. Acad. Sci. URSS. Ser. Math. [Izvestia Akad. Nauk SSSR] 1943, vol. 7, pp. 17–34 (in Russian).
  3. Vinogradov I. M. New approach to the estimation of a sum of values of χ(p + k). Izvestiya Akad. Nauk SSSR. Ser. Mat., 1952, vol. 16, pp. 197–210 (in Russian).
  4. Vinogradov I. M. Improvement of an estimate for the sum of the values χ(p+k). Izvestiya Akad. Nauk SSSR. Ser. Mat., 1953, vol 17, pp. 285–290 (in Russian).
  5. Vinogradov I. M. An estimate for a certain sum extended over the primes of an arithmetic progression. (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 1966, vol 30, no. 3, pp. 481–496 (in Russian).
  6. Karatsuba A. A. Sums of characters, and primitive roots, in finite fields. Doklady Akademii Nauk SSSR, 1968, vol 180, no. 6, pp. 1287–1289 (in Russian).
  7. Karatsuba A. A. Estimates of character sums. Math. USSR-Izv., 1970, vol. 4, no. 1, pp. 19–29.
  8. Karatsuba A. A. Sums of characters over prime numbers. Math. USSR-Izv., 1970, vol. 4, no. 2, pp. 303–326.
  9. Rakhmonov Z. Kh. On the distribution of values of Dirichlet characters. Rus. Math. Surv., 1986, vol. 41, no. 1, pp. 237–238. DOI: 10.1070/RM1986v041n01ABEH 003232.
  10. Rakhmonov Z. Kh. Estimation of the sum of characters with primes. Dokl. Akad. Nauk Tadzhik. SSR, 1986, vol 29, no. 1, pp. 16–20 (in Russian).
  11. Rakhmonov Z. Kh. On the distribution of the values of Dirichlet characters and their applications. Proc. Steklov Inst. Math., 1995, vol. 207, no. 6, pp. 263–272.
  12. Fridlander Dzh. B., Gong K., Shparlinskii I. E. Character sums over shifted primes. Math. Notes, 2010, vol. 88, iss. 3–4, pp. 585–598. DOI: 10.1134/S00014346 10090312.
  13. Rakhmonov Z. Kh. A theorem on the mean value of ψ(x, χ) and its applications. Russian Academy of Sciences. Izvestiya Mathematics, 1994, vol. 43, no. 1, pp. 49–64. DOI: 1070/IM1994v043n01ABEH001558.
  14. Rakhmonov Z. Kh. A theorem on the mean-value of Chebyshev functions. Russian Academy of Sciences. Izvestiya Mathematics, 1995, vol. 44, no. 3, pp. 555–569. DOI 1070/IM1995v044n03ABEH001613.
  15. Vinogradov A. I. On numbers with small prime divisors. Dokl. Akad. Nauk SSSR, 1956, vol 109, no. 4, pp. 683–686 (in Russian).
  16. Burgess D. A. On character sum estimate with r = 3. J. London Math. Soc., 1986, vol. 33, no. 2, pp. 219–226. DOI: 10.1112/jlms/s2-33.2.219. 
Short text (in English):
download
(downloads: 67)
  • 958 reads