RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Chebyshevskii Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

This article is cited in 1 scientific paper (total in 1 paper)

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
\begin{equation} 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} \end{equation}
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 . 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.

Full text: PDF file (758 kB)
References: PDF file   HTML file
UDC: 511.524
Received: 09.12.2015
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}


Linking options:
  • http://mi.mathnet.ru/eng/cheb464
  • http://mi.mathnet.ru/eng/cheb/v17/i1/p201

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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  mathnet  crossref  crossref  isi  elib
  • Number of views:
    This page:168
    Full text:61
    References:49

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020