Complete rational arithmetic sums

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• 作者：V. N. Chubarikov
• 刊名：Moscow University Mathematics Bulletin
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：71
• 期：1
• 页码：43-44
• 全文大小：457 KB
• 参考文献：1.I. M. Vinogradov, The Method of Trigonometric Sums in the Number Theory (Nauka, Moscow, 1980).[in Russian].MATH
  2.L.-K. Hua, The Method of Trigonometric Sums and its Applications to Number Theory (Mir, Moscow, 1964).[in Russian].
  3.J. R. Chen, “On Professor Hua’s Estimate on Exponential Sums,” Acta Sci. Sinica 20 (6), 711 (1977).MATH
  4.V. N. Chubarikov, “Multiple Trigonometric Sums and Multiple Integrals,” Matem. Zametki 20 (1), 61 (1976).MathSciNet MATH
  5.G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, Theory of Multiple Trigonometric Sum,s. (Nauka, Moscow, 1987).[in Russian].MATH
  6.G. I. Arkhipov, V. A. Sadovnichii, V. N. Chubarikov, Lectures in Mathematical Analysis (Drofa, Moscow, 2006).[in Russian].
  
• 作者单位：V. N. Chubarikov (1)
  
  1. Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Russian Library of Science
  
• 出版者：Allerton Press, Inc. distributed exclusively by Springer Science+Business Media LLC
• ISSN：1934-8444

文摘

Let {itq} > 1 be an integer number, \(f\left( x \right) = {a_n}{x^n} + \ldots + {a_1}x + {a_0}\) be a polynomial with integer coefficients, and ({ita}{in{itn}}, . . . ,{ita}{in1},{itq}) = 1. The following estimate is valid: \(\left| {S\left( {\frac{{f\left( x \right)}}{q}} \right)} \right| = \left| {\sum\limits_{x = 1}^q \rho \left( {\frac{{f\left( x \right)}}{q}} \right)} \right| \ll {q^{1 - 1/n}}\), where \(\rho \left( t \right) = 0,5 - \left\{ t \right\}\).