Geodesic
Sums of values of nonprincipal characters over a sequence of shifted primes
Informatics and Automation, Analytic number theory, Tome 299 (2017), pp. 234-260.

Voir la notice de l'article provenant de la source Math-Net.Ru

For a nonprincipal character $\chi $ modulo $D$, we prove a nontrivial estimate of the form $\sum _{n\le x}\Lambda (n)\chi (n-l)\ll x\exp \{-0.6\sqrt {\ln D}\}$ for the sum of values of $\chi $ over a sequence of shifted primes in the case when $x\ge D^{1/2+\varepsilon }$, $(l,D)=1$, and the modulus of the primitive character generated by $\chi $ is a cube-free number.
Keywords: Dirichlet character, shifted primes, short character sum, exponential sum over primes. 
@article{TRSPY_2017_299_a14,
     author = {Z. Kh. Rakhmonov},
     title = {Sums of values of nonprincipal characters over a sequence of shifted primes},
     journal = {Informatics and Automation},
     pages = {234--260},
     publisher = {mathdoc},
     volume = {299},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2017_299_a14/}
}
TY  - JOUR
AU  - Z. Kh. Rakhmonov
TI  - Sums of values of nonprincipal characters over a sequence of shifted primes
JO  - Informatics and Automation
PY  - 2017
SP  - 234
EP  - 260
VL  - 299
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2017_299_a14/
LA  - ru
ID  - TRSPY_2017_299_a14
ER  - 
%0 Journal Article
%A Z. Kh. Rakhmonov
%T Sums of values of nonprincipal characters over a sequence of shifted primes
%J Informatics and Automation
%D 2017
%P 234-260
%V 299
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2017_299_a14/
%G ru
%F TRSPY_2017_299_a14
Z. Kh. Rakhmonov. Sums of values of nonprincipal characters over a sequence of shifted primes. Informatics and Automation, Analytic number theory, Tome 299 (2017), pp. 234-260. http://geodesic.mathdoc.fr/item/TRSPY_2017_299_a14/

[1] Burgess D.A., “On character sums and $L$-series”, Proc. London Math. Soc. Ser. 3, 12 (1962), 193–206 | DOI | MR | Zbl

[2] Burgess D.A., “On character sums and $L$-series. II”, Proc. London Math. Soc. Ser. 3, 13 (1963), 524–536 | DOI | MR | Zbl

[3] J. B. Friedlander, K. Gong, and I. E. Shparlinskii, “Character sums over shifted primes”, Math. Notes, 88 (2010), 585–598 | DOI | DOI | MR | Zbl

[4] A. A. Karatsuba, “Character sums and primitive roots in finite fields”, Sov. Math., Dokl., 9 (1968), 755–757 | MR | Zbl

[5] A. A. Karatsuba, “Estimates of character sums”, Math. USSR, Izv., 4:1 (1970), 19–29 | DOI | MR

[6] A. A. Karatsuba, “Sums of characters over prime numbers”, Math. USSR, Izv., 4:2 (1970), 303–326 | DOI | MR

[7] A. A. Karatsuba, “On sums of characters with primes”, Sov. Math., Dokl., 11 (1970), 135–137 | MR | Zbl

[8] A. A. Karatsuba, “The distribution of products of shifted primes in arithmetical progressions”, Sov. Math., Dokl., 11 (1970), 707–711 | MR | Zbl

[9] A. A. Karatsuba, “Sums of characters with prime numbers in an arithmetic progression”, Math. USSR, Izv., 5:3 (1971), 485–501 | DOI | MR | Zbl

[10] A. A. Karatsuba, “Sums of characters in sequences of shifted prime numbers, with applications”, Math. Notes, 17 (1975), 91–93 | DOI | MR | Zbl

[11] A. A. Karatsuba, “Some problems of contemporary analytic number theory”, Math. Notes, 17 (1975), 195–199 | DOI | MR | MR | Zbl

[12] A. A. Karatsuba, “On the distribution of values of nonprincipal characters”, Proc. Steklov Inst. Math., 142 (1979), 165–174 | MR | Zbl

[13] A. A. Karatsuba, “Sums of Legendre symbols of polynomials of second degree over prime numbers”, Math. USSR, Izv., 12:2 (1978), 299–308 | DOI | MR | Zbl | Zbl

[14] A. A. Karatsuba, Basic Analytic Number Theory, Springer, Berlin, 1993 | MR | MR | Zbl

[15] A. A. Karatsuba, “Arithmetic problems in the theory of Dirichlet characters”, Russ. Math. Surv., 63 (2008), 641–690 | DOI | DOI | MR | Zbl

[16] Yu. V. Linnik, “Recent works of I. M. Vinogradov”, Proc. Steklov Inst. Math., 132 (1973), 25–28 | MR | Zbl

[17] Z. Kh. Rakhmonov, “On the distribution of values of Dirichlet characters”, Russ. Math. Surv., 41:1 (1986), 237–238 | DOI | MR | Zbl | Zbl

[18] Z. Kh. Rakhmonov, “On an estimate for a character sum over primes”, Dokl. Akad. Nauk Tadzhik. SSR, 29:1 (1986), 16–20 | MR | Zbl

[19] Z. Kh. Rakhmonov, “On the least Goldbach number in an arithmetic progression”, Izv. Akad. Nauk Tadzhik. SSR, Otd. Fiz.-Mat. Khim. Geol. Nauk, 1986, no. 2, 103–106 | MR | Zbl

[20] Z. Kh. Rakhmonov, “Theorem on the mean value of $\psi (x,\chi )$ and its applications”, Russ. Acad. Sci., Izv. Math., 43:1 (1994), 49–64 | MR | Zbl

[21] Z. Kh. Rakhmonov, “On the distribution of values of Dirichlet characters and their applications”, Proc. Steklov Inst. Math., 207 (1995), 263–272 | MR | Zbl

[22] Z. Kh. Rakhmonov, “Distribution of values of Dirichlet characters in the sequence of shifted primes”, Dokl. Akad. Nauk Resp. Tadzhikistan, 56:1 (2013), 5–9

[23] Z. Kh. Rakhmonov, “Distribution of values of Dirichlet characters in a sequence of shifted primes”, Izv. Saratov. Univ., Nov. Ser., Mat. Mekh. Inf., 13:4-2 (2013), 113–117 | Zbl

[24] Z. Kh. Rakhmonov, “Sums of characters over prime numbers”, Chebyshev. Sb., 15:2 (2014), 73–100

[25] Z. Kh. Rakhmonov and Sh. Kh. Mirzorakhimov, “Sums of characters modulo a cubefree at shifted primes”, Chebyshev. Sb., 17:1 (2016), 201–216 | MR

[26] A. I. Vinogradov, “On numbers with small prime divisors”, Dokl. Akad. Nauk SSSR, 109:4 (1956), 683–686 | Zbl

[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 (1938), 311–319 | Zbl

[28] I. M. Vinogradov, “An improvement of the estimation of sums with primes”, Izv. Akad. Nauk SSSR, Ser. Mat., 7:1 (1943), 17–34

[29] I. M. Vinogradov, “New approach to the estimation of a sum of values of $\chi (p+k)$”, Izv. Akad. Nauk SSSR, Ser. Mat., 16:3 (1952), 197–210 | Zbl

[30] I. M. Vinogradov, “Improvement of an estimate for the sum of the values $\chi (p+k)$”, Izv. Akad. Nauk SSSR, Ser. Mat., 17:4 (1953), 285–290 | Zbl

[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 (1966), 481–496 | Zbl