Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-08T06:33:50.520Z Has data issue: false hasContentIssue false
  • English
  • Français

SUMS OF MULTIPLICATIVE FUNCTIONS OVER A BEATTY SEQUENCE

Part of: Sequences and sets Forms and linear algebraic groups

Published online by Cambridge University Press:  01 October 2008

AHMET M. GÜLOĞLU  and
C. WESLEY NEVANS
AHMET M. GÜLOĞLU*
Affiliation:
Department of Mathematics, Bilkent University, Bilkent 06800, Ankara, Turkey (email: [email protected])
C. WESLEY NEVANS
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study sums involving multiplicative functions that take values over a nonhomogenous Beatty sequence. We then apply our result in a few special cases to obtain asymptotic formulas for quantities such as the number of integers in a Beatty sequence that are representable as a sum of two squares up to a given magnitude.

Keywords

sums of multiplicative functionsBeatty sequences

MSC classification

Secondary: 11E25: Sums of squares and representations by other particular quadratic forms 11B83: Special sequences and polynomials
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 2 , October 2008 , pp. 327 - 334
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Bachman, G., ‘On exponential sums with multiplicative coefficients, II’, Acta Arith. 106(1) (2003), 41–57.Google Scholar
[2]Bugeaud, Y., Approximation by Algebraic Numbers, Cambridge Tracts in Mathematics, 160 (Cambridge University Press, Cambridge, 2004).Google Scholar
[3]Gegenbauer, L., ‘Asymptotische Gesetze der Zahlentheori’, Denkschirften Akad. Wien 49(1) (1885), 37–80.Google Scholar
[4]Hardy, G. and Wright, E., An Introduction to the Theory of Numbers, 5th edn (Oxford University Press, New York, 2005).Google Scholar
[5]Iwaniec, H. and Kowalski, E., Analytic Number Theory, American Mathematical Society Colloquium Publications, 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
[6]Khinchin, A. Y., ‘Zur metrischen Theorie der diophantischen approximationen’, Math. Z. 24(4) (1926), 706–714.Google Scholar
[7]Kuipers, L. and Niederreiter, H., Uniform Distribution of Sequences, Pure and Applied Mathematics (Wiley-Interscience, New York, 1974).Google Scholar
[8]Montgomery, H. L. and Vaughan, R. C., ‘Exponential sums with multiplicative coefficients’, Invent. Math. 43(1) (1977), 69–82.Google Scholar
[9]Ramanujan, S., ‘Some formulae in the analytic theory of numbers’, Messenger Math. 45 (1916), 81–84.Google Scholar
[10]Roth, K. F., ‘Rational approximations to algebraic numbers’, Mathematika 2 (1955), 1–20; ‘Corrigendum’, Mathematika 2 (1955), 168.Google Scholar
[11]Schmidt, W. M., Diophantine Approximation, Lecture Notes in Mathematics, 785 (Springer, Berlin, 1980).Google Scholar
[12]Serre, J. P., ‘Divisibilité de certaines fonctions arithmétiques’, Enseign. Math. (2) 22(3–4) (1976), 227–260.Google Scholar
[13]Shanks, D., ‘The second-order term in the asymptotic expansion of B(x)’, Math. Comp. 18(85) (1964), 75–86.Google Scholar
[14]Smith, R. A., ‘An error term of Ramanujan’, J. Number Theory 2 (1970), 91–96.CrossRefGoogle Scholar
[15]Vinogradov, I. M., The Method of Trigonometrical Sums in the Theory of Numbers (Dover Publications, Inc., Mineola, NY, 2004).Google Scholar