Functiones et Approximatio Commentarii Mathematici

On multiple exponential sums and their applications

H.-Q Liu

Full-text: Open access

PDF File (414 KB)

Abstract

We prove new estimates for the remainder terms in the known asymptotic formulas for three famous problems, by using the contemporary bounds for triple exponential sums.

Article information

Source
Funct. Approx. Comment. Math., Volume 45, Number 2 (2011), 155-163.

Dates
First available in Project Euclid: 12 December 2011

Permanent link to this document
https://projecteuclid.org/euclid.facm/1323705812

Digital Object Identifier
doi:10.7169/facm/1323705812

Mathematical Reviews number (MathSciNet)
MR2895153

Zentralblatt MATH identifier
1278.11079

Subjects
Primary: 11L07: Estimates on exponential sums

Keywords
multiple exponential sums direct factors of abelian groups squarefree divisor problem primitive circle problem

Citation

Liu, H.-Q. On multiple exponential sums and their applications. Funct. Approx. Comment. Math. 45 (2011), no. 2, 155--163. doi:10.7169/facm/1323705812. https://projecteuclid.org/euclid.facm/1323705812


Export citation

References

  • R. C. Baker, The squarefree divisor problem(II), Quart. J. Math. (Oxford Series) 47(2) (1996), 133–147.
    Mathematical Reviews (MathSciNet): MR1397933
    Zentralblatt MATH: 0866.11053
    Digital Object Identifier: doi:10.1093/qmath/47.2.133
  • E. Bombieri and H. Iwaniec, On the order of $\varsigma (1/2+it)$, Ann. Scuola Norm. Sup. Pisa 13 (1986), 449–473.
  • E. Cohen, On the average number of direct factors of a finite abelian group, Acta Arith. 6 (1960), 159–173.
    Mathematical Reviews (MathSciNet): MR118764
  • E. Fouvry and H. Iwaniec, Exponential sums with monomials, J. Number Theory 33 (1989), 311–333.
    Mathematical Reviews (MathSciNet): MR1027058
    Zentralblatt MATH: 0687.10028
    Digital Object Identifier: doi:10.1016/0022-314X(89)90067-X
  • D. R. Heath-Brown, The Pjateckii-Sapiro prime number theorem, ibid, 16 (1983), 242–266.
    Mathematical Reviews (MathSciNet): MR698168
    Digital Object Identifier: doi:10.1016/0022-314X(83)90044-6
  • C. H. Jia, The distribution of squarefree numbers, Sci. China Ser. A, 36, no.2, (1993), 154–169.
    Mathematical Reviews (MathSciNet): MR1223084
  • E. Krätzel, On the average number of direct factors of a finite abelian group, Acta Arith. 51 (1988), 369–379.
    Mathematical Reviews (MathSciNet): MR971087
  • H.-Q. Liu, On some divisor problems, ibid, 68 (1994), 193–200.
    Mathematical Reviews (MathSciNet): MR1305200
    Zentralblatt MATH: 0817.11043
  • H.-Q. Liu, Divisor problems of 4 and 3 dimensions, ibid, 73 (1995), 249–269.
    Mathematical Reviews (MathSciNet): MR1364462
    Zentralblatt MATH: 0846.11056
  • H.-Q. Liu, On a fundamental result in van der Corput's method of exponential sums, ibid, 90 (1999), 357–370.
    Mathematical Reviews (MathSciNet): MR1723675
    Zentralblatt MATH: 0934.11039
  • H.-Q. Liu, Exponential sums and the abelian group problem, Functiones et Approximitio 42(2) (2010), 113–129.
    Mathematical Reviews (MathSciNet): MR2674533
    Project Euclid: euclid.facm/1277811635
  • H.-Q. Liu, On the average number of unitary factors of finite abelian groups, ibid, 43(1) (2010), 7–14.
    Mathematical Reviews (MathSciNet): MR2683569
    Project Euclid: euclid.facm/1285679141
  • H.-Q. Liu and J. Wu, On the average number of direct factors of a finite abelian group, preprint, 1999.
    Mathematical Reviews (MathSciNet): MR1796804
  • H.-Q. Liu and J. Wu, Numbers with a large prime factor, Acta Arith. 89 (1999), 163–187.
    Mathematical Reviews (MathSciNet): MR1691896
    Zentralblatt MATH: 0937.11038
  • H. Menser, Vierdimensionale Gitterpunktprobleme I,II, Forschungsergebnisse, FSU, Jena, N/89/38, N/89/02.
  • H. Menser and R. Seibold, On the average number of direct factors of a finite abelian group, Monatsh. Math. 110 (1990), 63–72.
    Mathematical Reviews (MathSciNet): MR1072729
  • O. Robert and P. Sargos, Three dimensional exponential sums with monomials, J. Reine Angew. Math. 591 (2006), 1–20.
    Mathematical Reviews (MathSciNet): MR2212877
  • P. Sargos ang J. Wu, Multiple exponential sums with monomials and their applications in number theory, Acta Math. Hungar. 87 (2000), 333–354.
    Mathematical Reviews (MathSciNet): MR1771211
    Zentralblatt MATH: 0963.11045
    Digital Object Identifier: doi:10.1023/A:1006777803163
  • M. Vogts, Many-dimensional generalized divisor problems, Math. Nachr. 124 (1985), 103–124.
    Mathematical Reviews (MathSciNet): MR827893
    Zentralblatt MATH: 0595.10034
    Digital Object Identifier: doi:10.1002/mana.19851240108
  • J. Wu, On the average number of direct factors of a finite abelian group, Monatsh. Math. 131 (2000), 79–89.
    Mathematical Reviews (MathSciNet): MR1796804
  • J. Wu, On the average number of unitary factors of finite abelian groups, Acta Arith. 84 (1998), 17–29.
    Mathematical Reviews (MathSciNet): MR1613290
    Zentralblatt MATH: 0899.11048
  • J. Wu, On the distribution of square-full integers, Arch. Math. (Basel) 77(3) (2001), 233–240.
    Mathematical Reviews (MathSciNet): MR1865864
    Zentralblatt MATH: 1003.11043
    Digital Object Identifier: doi:10.1007/PL00000486
  • J. Wu, On the primitive circle problem, Monatsh. Math. 135 (2002), 69–81.
    Mathematical Reviews (MathSciNet): MR1894296
  • G. Yu, On the average number of direct factors of finite abelian groups, Acta Math. Sinica 37 (1994), 663–670 (in Chinese).
    Mathematical Reviews (MathSciNet): MR1311525
  • W. Zhai and X. Cao, On the average number of direct factors of finite abelian groups, Acta Arith. 82 (1997), 45–55.
    Mathematical Reviews (MathSciNet): MR1475765

Adam Mickiewicz University, Faculty of Mathematics and Computer Science

Functiones et Approximatio Commentarii Mathematici

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more