Sums of characters over prime numbers

Z. Kh. Rakhmonov

Čebyševskij sbornik, Tome 15 (2014) no. 2, pp. 73-100.

Résumé

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 $p-l$, $(l,q)=1$, $p\le x$. This estimate is nontrivial for $ x \ge q^{\frac{5}{6}+\varepsilon}$ and refines the estimate obtained by J. B. Friedlander, K. Gong, I. E. Shparlinskii. Their estimate holds provided that $x\ge q^{\frac{8}{9}+\varepsilon}$. Bibliography: 20 titles.

Détail

Keywords: Dirichlet character, shifted primes, short sums of characters, exponential sums over primes.

Bibliographie

[1] Vinogradov I. M., “Raspredelenie kvadratichnykh vychetov i nevychetov vida $p+k$ po prostomu modulyu”, Matematicheskii sbornik, 3:45 (1938), 311–320

[2] Vinogradov I. M., “Utochnenie metoda otsenki summ s prostymi chislami”, Izvestiya AN SSSR. Ser. Mat., 7 (1943), 17–34

[3] Vinogradov I. M., “Novyi podkhod k otsenke summy znachenii $\chi (p+k)$”, Izvestiya AN SSSR. Ser. Mat., 16 (1952), 197–210 | MR | Zbl

[4] Vinogradov I. M., “Uluchshenie otsenki dlya summy znachenii $\chi (p+k)$”, Izvestiya AN SSSR. Ser. Mat., 17 (1953), 285–290 | MR | Zbl

[5] Vinogradov I. M., “Otsenka odnoi summy, rasprostranennoi na prostye chisla arifmeticheskoi progressii”, Izvestiya AN SSSR. Ser. Mat., 30 (1966), 481–496 | MR | Zbl

[6] Karatsuba A. A., “Summy kharakterov i pervoobraznye korni v konechnykh polyakh”, Doklady AN SSSR, 180:6 (1968), 1287–1289 | Zbl

[7] Karatsuba A. A., “Ob otsenkakh summ kharakterov”, Izvestiya AN SSSR. Ser. Mat., 34 (1970), 20–30

[8] Karatsuba A. A., “Summy kharakterov s prostymi chislami”, Izvestiya AN SSSR. Ser. Mat., 34 (1970), 299–321

[9] Rakhmonov Z. Kh., “O raspredelenii znachenii kharakterov Dirikhle”, UMN, 41:1 (1986), 201–202 | MR | Zbl

[10] Rakhmonov Z. Kh., “Ob otsenke summy kharakterov s prostymi chislami”, DAN Tadzhikskii SSR, 29:1 (1986), 16–20 | MR | Zbl

[11] Rakhmonov Z. Kh., “O raspredelenii znachenii kharakterov Dirikhle i ikh prilozheniya”, Trudy Matematicheskogo instituta RAN, 207, 1994, 286–296 | MR | Zbl

[12] Rakhmonov Z. Kh., “O raspredelenii znachenii kharakterov Dirikhle v posledovatelnosti sdvinutykh prostykh chisel”, Doklady AN Respubliki Tadzhikistan, 56:1 (2013), 5–9

[13] Rakhmonov Z. Kh., “Raspredelenie znachenii kharakterov Dirikhle v posledovatelnosti sdvinutykh prostykh chisel”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 13:4(2) (2013), 113–117

[14] Dzh. B. Fridlander, K. Gong, I. E. Shparlinskii, “Summy znachenii kharakterov na sdvinutykh prostykh chislakh”, Mat. zametki, 88:4 (2010), 605–619 | DOI | MR

[15] Rakhmonov Z. Kh., “Teorema o srednem znachenii $\psi (x,\chi)$ i ee prilozheniya”, Izvestiya RAN. Ser. Mat., 57:4 (1993), 55–71 | MR | Zbl

[16] Rakhmonov Z. Kh., “Teorema o srednem znachenii funktsii Chebysheva”, Izvestiya RAN. Ser. Mat., 58:3 (1994), 127–139 | MR | Zbl

[17] Vinogradov A. I., “O chislakh s malymi prostymi delitelyami”, DAN SSSR, 109:4 (1956), 683–686 | MR | Zbl

[18] Burgess D. A., “The character sum estimate with $r = 3$”, J. London Math. Soc., 33 (1986), 219–226 | DOI | MR | Zbl

[19] Karatsuba A. A., Osnovy analiticheskoi teorii chisel, Nauka, M., 1983 | MR

[20] Mardzhanishvili K. K., “Otsenka odnoi arifmeticheskoi summy”, DAN SSSR, 22:7 (1939), 391–393