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New estimates of the remainder in an asymptotic formula in the multidimensional Dirichlet divisor problem

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Abstract

We obtain a new value of the Karatsuba constant in the multidimensional Dirichlet divisor problem. We also find a new value of the exponent of the main parameter in the estimate of the mean value of the remainder in a given asymptotics. The proof of the main statements is based on the derivation of a new estimate of the Carleson abscissa in the theory of the Riemann zeta function.

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References

  1. P. G. L. Dirichlet, “Über die Bestimmung der mittleren Werthe in der Zahlentheorie,” Abh. Acad. Wiss. Berlin, 69–83 (1849); in Werke, Bd. 2, pp. 49–66.

  2. G. Voronoï, “Sur un problème du calcul des fonctions asymptotiques,” Reine Angew. Math. 126, 241–282 (1903).

    Article  MATH  Google Scholar 

  3. E. Landau, “Über die Anzahl der Gitterpunkte in gewissen Bereichen,” G ött. Nachr., 687–770 (1912).

  4. G. H. Hardy and J. E. Littlewood, “The approximate functional equation in the theory of the zeta-function, with applications to the divisor problems of Dirichlet and Piltz,” Proc. London Math. Soc. (2) 21, 39–74 (1922).

    Article  Google Scholar 

  5. J. G. van der Corput, “Versch ärfung der Absch ätzung beim Teilerproblem,” Math. Ann. 87(1–2), 39–65 (1922).

    Article  MathSciNet  Google Scholar 

  6. K. C. Tong, “On diviser problems,” Acta Math. Sinica (Chin. Ser.) 2, 258–266 (1952).

    Google Scholar 

  7. A. Walfisz, “Über zwei Gitterpunktprobleme,” Math. Ann. 95(1), 69–83 (1926).

    Article  MathSciNet  Google Scholar 

  8. F. V. Atkinson, “A divisor problem,” Quart. J.Math., Oxford Ser. 12(1), 193–200 (1941).

    Article  MathSciNet  Google Scholar 

  9. T. Chih, “The Dirichlet’s divisor problem,” Sci. Rep. Nat. Tsing Hua Univ. Ser. A 5, 402–427 (1950).

    MathSciNet  Google Scholar 

  10. H.-E. Richert, “Versh ärfung der Absch ärzung beim Dirichletschen Teilerproblem,” Math. Z. 58(1), 204–218 (1953).

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Chen, “On the divisor problem for d 3(n),” Sci. Sinica 14, 19–29 (1965).

    MathSciNet  MATH  Google Scholar 

  12. A. A. Karatsuba, “A uniform estimate for the remainder term in Dirichlet’s problem of divisors,” Izv. Akad. Nauk SSSR Ser. Mat. 36(3), 475–483 (1972).

    MathSciNet  MATH  Google Scholar 

  13. G. A. Kolesnik, “An improvement of the remainder term in the divisor problem,” Mat. Zametki 6(5), 545–554 (1969) [Math. Notes 6 (5), 784–791 (1969)].

    MathSciNet  Google Scholar 

  14. A. Ivić, “Some recent results on the Riemann zeta-function,” in Théorie des nombres, Quebec, PQ, 1987 (de Gruyter, Berlin, 1989), pp. 424–440.

    Google Scholar 

  15. A. Ivić and M. Ouellet, “Some new estimates in the Dirichlet divisor problem,” Acta Arith. 52 (No. 3), 241–253 (1989).

    MathSciNet  MATH  Google Scholar 

  16. E. E. Bayadilov, “On the divisor problem for values of a ternary cubic form,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 1, 58–60 (1999) [Moscow Univ. Math. Bull. 54 (1), 43–45 (1999)].

  17. K. Ford, “Vinogradov’s integral and bounds for the Riemann zeta function,” Proc. London Math. Soc. (3) 85(No. 3), 565–633 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  18. H.-E. Richert, “Einf ührung in die Theorie der starken Rieszschen Summierbarkeit von Dirichletreihen,” Nachr. Akad.Wiss. Göttingen Math.-Phys. Kl. II 1960, 17–75 (1960).

    MathSciNet  Google Scholar 

  19. A. A. Karatsuba, “Estimates of trigonometric sums by the method of I. M. Vinogradov, and their applications,” in Trudy Mat. Inst. Steklov Collection of articles dedicated to Academician I. M. Vinogradov on his eightieth birthday (Nauka, Moscow, 1971), Vol. 112, pp. 241–255 [in Russian].

    Google Scholar 

  20. A. Fujii, “On the problem of divisors,” Acta Arith. 31(4), 355–360 (1976).

    MathSciNet  MATH  Google Scholar 

  21. E. I. Panteleeva, “Dirichlet divisor problem in number fields,” Mat. Zametki 44(4), 494–505 (1988) [Math. Notes 44 (4), 750–757 (1988)].

    MathSciNet  MATH  Google Scholar 

  22. E. C. Titchmarsh, The Theory of the Riemann Zeta Function (Oxford, 1951; Inostr. Lit., Moscow, 1953).

  23. A. Ivić, The Riemann Zeta-Function. The Theory of the Riemann Zeta-Function with Applications, in Wiley-Intersci. Publ. (John Wiley & Sons, New York, 1985).

    Google Scholar 

  24. E. C. Titchmarsh, “On the remainder in the formula for N(T), the number of zeros of ζ(s) in the strip 0 < t < T,” Proc. London Math. Soc. (2) 27(1), 449–458 (1928).

    Article  Google Scholar 

  25. G. I. Arkhipov and K. Buriev, “Refinement of estimates for the riemann zeta-function in a neighbourhood of the line Re s = 1,” Integral Transform. Spec. Funct. 1(1), 1–7 (1993).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    O. V. Kolpakova

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  1. O. V. Kolpakova
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Original Russian Text © O. V. Kolpakova, 2011, published in Matematicheskie Zametki, 2011, Vol. 89, No. 4, pp. 530–546.

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Kolpakova, O.V. New estimates of the remainder in an asymptotic formula in the multidimensional Dirichlet divisor problem. Math Notes 89, 504–518 (2011). https://doi.org/10.1134/S0001434611030229

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