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 Chebyshevskii Sb., 2016, Volume 17, Issue 1, Pages 201–216 (Mi cheb464)

Sums of characters modulo a cubefree at shifted primes

Z. Kh. Rakhmonov, Sh. Kh. Mirzorakhimov

Institute of Mathematics, Academy of Sciences of Republic of Tajikistan, Dushanbe

Abstract: Vinogradov's method of estimation of exponential sums over primes allowed him to solve the number of arithmetic problems with primes. One of them is a problem of distribution of the values of non-principal character on the sequence of shifted primes. In 1938 he proved that if $q$ is an odd prime, $(l, q)=1$, $\chi (a)$ is non-principal character modulo $q$, then
$$T(\chi )=\sum_{p\le x}\chi (p-l)\ll x^{1+\varepsilon} (\sqrt{\frac{1}{q}+\frac{q}{x}} +x^{-\frac{1}{6}}). \tag{IMV}$$
This estimate is non-trivial when $x\gg q^{1+\varepsilon}$ and an asymptotic formula for the the number of quadratic residues (non-residues) modulo $q$ of the form $p-l$, $p\le x$ follows from it. Later in 1953, I. M. Vinogradov obtained a non-trivial estimate of $T(\chi )$ when $x\ge q^{0,75+\varepsilon}$, $q$ is a prime. It was a surprising result. In fact, $T(\chi )$ can be represented as a sum over zeroes of correspondent Dirichlet $L$ — function; So a non-trivial estimate of $T(\chi )$ is obtained only for $x \ge q^{1+\varepsilon}$ provided that the extended Riemann hypothesis is true.
In 1968 A. A. Karatsuba found a method that allowed him to obtain non-trivial estimate of short sums of characters in finite fields with fixed degree. In 1970 using the modification of his technique coupled with Vinogradov's method he proved that: if $q$ is a prime number, $\chi$ is non-principal character modulo $q$ and $x\ge q^{\frac{1}{2}+\varepsilon}$, then the following estimate is true
$$T(\chi )\ll xq^{-\frac{1}{1024}\varepsilon^2}.$$

In 1985 Z. Kh. Rakhmonov generalized the estimate (IMV) for the case of composite modulo and proved: let $D$ is a sufficiently large positive integer, $\chi$ is a non-principal character modulo $D$, $\chi_q$ is primitive character generated by character $\chi$, then
$$T(\chi )\le x\ln^5x (\sqrt{\frac{1}{q}+\frac{q}{x}\tau^2(q_1)} +x^{-\frac{1}{6}}\tau (q_1)), \qquad q_1={\genfrac {0pt} {p\backslash D}{p\not\backslash q}}p.$$
If a character $\chi$ coincides with it generating primitive character $\chi_q$, then the last estimate is non-trivial for $x>q(\ln q)^{13}$.
In 2010 ã. J. B. Friedlander, K. Gong, I. E. Shparlinski showed that a non-trivial estimate of the sum $T(\chi_q )$ exists for composite $q$ when $x$ — length of the sum, is of smaller order than $q$. They proved: for a primitive character $\chi_q$ and an arbitrary $\varepsilon >0$ there exists such $\delta >0$ that for all $x\ge q^{\frac{8}{9}+\varepsilon}$ the following estimate holds:
$$T(\chi_q )\ll xq^{-\delta}.$$
In 2013 Z. Kh. Rakhmonov obtained a non-trivial estimate of $T(\chi_q)$ for the composite modulo $q$ and primitive character $\chi_q$ when $x\ge q^{\frac{5}{6}+\varepsilon}$.
In this paper the theorem about the estimate of the sum $T(\chi_q)$ is proved for cubefree modulo $q$. It is non-trivial when $x\ge q^{\frac{5}{6}+\varepsilon}$.
Bibliography: 15 titles.

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

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UDC: 511.524
Accepted:10.03.2016

Citation: Z. Kh. Rakhmonov, Sh. Kh. Mirzorakhimov, “Sums of characters modulo a cubefree at shifted primes”, Chebyshevskii Sb., 17:1 (2016), 201–216

Citation in format AMSBIB
\Bibitem{RakMir16} \by Z.~Kh.~Rakhmonov, Sh.~Kh.~Mirzorakhimov \paper Sums of characters modulo a cubefree at shifted primes \jour Chebyshevskii Sb. \yr 2016 \vol 17 \issue 1 \pages 201--216 \mathnet{http://mi.mathnet.ru/cheb464} \elib{http://elibrary.ru/item.asp?id=25795083} 

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This publication is cited in the following articles:
1. Z. Kh. Rakhmonov, “Sums of values of nonprincipal characters over a sequence of shifted primes”, Proc. Steklov Inst. Math., 299 (2017), 219–245
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