Geodesic
The rate of convergence of the average value of the full rational arithmetic sums
Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 303-318

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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. 
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     author = {V. N. Chubarikov},
     title = {The rate of convergence of the average value of the full rational arithmetic sums},
     journal = {\v{C}eby\v{s}evskij sbornik},
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V. N. Chubarikov. The rate of convergence of the average value of the full rational arithmetic sums. Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 303-318. http://geodesic.mathdoc.fr/item/CHEB_2015_16_4_a15/