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Chebyshevskii Sb., 2015, Volume 16, Issue 4, Pages 303–318 (Mi cheb447)  

This article is cited in 6 scientific papers (total in 6 papers)

The rate of convergence of the average value of the full rational arithmetic sums

V. N. Chubarikov

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: In this paper the exact value of a index of convergence for the mean-value of the complete rational arithmetical for the arithmetical function, satisfying the functional equation of Gaussian type, is found. In particular, the Bernoulli's polynomials satisfy for this functional equation.
A similar result holds for the complete rational trigonometric sums (Hua Loo-keng, 1952). The deduction of the main result of the paper leads of the elementary method. We owe to I. M. Vinogradov for the demonstration of fruitful results and profit of it.
The complete rational arithmetic sums are the analogue the oscillatory integral of a periodic function, for example, trigonometric functions. In 1978 similar results for the exact value of the index of convergence of the trigonometric integral were obtained (G. I. Arkhipov, A. A. Karatsuba, V. N. Chubarikov).
In nowadays for a multivariate problem there are successful to get only upper and lower estimates for the index of convergence of appropriate sums and integrals.
Bibliography: 19 titles.

Keywords: the Gauss theorem of a multiplication for the Euler gamma-function, complete rational arithmetical sums, a functional equation on a complete system of residues by modulo of natural number, the Bernoulli polynomials.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00835


Full text: PDF file (271 kB)
References: PDF file   HTML file
UDC: 511.9
Received: 01.12.2015

Citation: V. N. Chubarikov, “The rate of convergence of the average value of the full rational arithmetic sums”, Chebyshevskii Sb., 16:4 (2015), 303–318

Citation in format AMSBIB
\Bibitem{Chu15}
\by V.~N.~Chubarikov
\paper The rate of convergence of the average value of the full rational arithmetic sums
\jour Chebyshevskii Sb.
\yr 2015
\vol 16
\issue 4
\pages 303--318
\mathnet{http://mi.mathnet.ru/cheb447}
\elib{https://elibrary.ru/item.asp?id=25006105}


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    This publication is cited in the following articles:
    1. V. N. Chubarikov, “On an elementary version of I.M. Vinogradov's method”, Proc. Steklov Inst. Math., 296 (2017), 41–51  mathnet  crossref  crossref  mathscinet  isi  elib
    2. V. N. Chubarikov, “On complete rational arithmetic sums of polynomial values”, Proc. Steklov Inst. Math., 299 (2017), 50–55  mathnet  crossref  crossref  isi  elib  elib
    3. V. N. Chubarikov, M. L. Sharapova, “A cubature formula for periodic functions”, Moscow University Mathematics Bulletin, 72:6 (2017), 255–257  mathnet  crossref  mathscinet  isi
    4. L. G. Arkhipova, V. N. Chubarikov, “Convergence exponent of a singular series for a multi-dimensional problem”, Moscow University Mathematics Bulletin, Moscow University Mchanics Bulletin, 73:5 (2018), 207–209  mathnet  crossref  mathscinet  zmath  isi
    5. V. N. Chubarikov, “Obobschennye summy Gaussa i mnogochleny Bernulli”, Chebyshevskii sb., 20:1 (2019), 284–293  mathnet  crossref
    6. L. G. Arkhipova, V. N. Chubarikov, “O pokazatelyakh skhodimosti osobogo integrala i osobogo ryada odnoi mnogomernoi problemy”, Chebyshevskii sb., 20:4 (2019), 46–57  mathnet  crossref
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