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Monatshefte für Mathematik

, Volume 164, Issue 2, pp 157–170 | Cite as

Formulas for the number of gridlines

  • Anne-Maria Ernvall-Hytönen
  • Kaisa Matomäki
  • Pentti Haukkanen
  • Jorma K. Merikoski
Article

Abstract

Let l(n) be the number of lines through at least two points of an n × n rectangular grid. We prove recursive and asymptotic formulas for it using respectively combinatorial and number theoretic methods. We also study the ratio l(n)/l(n − 1). All this originates from Mustonen’s experimental results.

Keywords

Rectangular grid Lattice points Euler \({\phi}\)-function Asymptotic formulas Recursive formulas 

Mathematics Subject Classification (2000)

05A99 11B37 11N37 11P21 

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References

  1. 1.
    Apostol T.M.: Introduction to Analytic Number Theory. Springer, Berlin (1976) Third printing, 1986zbMATHGoogle Scholar
  2. 2.
    Bender E.A., Patashnik O., Rumsey H. Jr: Pizza slicing, phi’s and the Riemann hypothesis. Am. Math. Mon. 101, 307–317 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Codecà P.: A note on Euler’s \({\phi}\)-function. Ark. Mat. 19, 261–263 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Mitrinović D.S., Sándor J., Crstici B.: Handbook of Number Theory. Kluwer, Dordrecht (1996)Google Scholar
  5. 5.
    Mustonen, S.: On lines and their intersection points in a rectangular grid of points. http://www.survo.fi/papers/PointsInGrid.pdf
  6. 6.
    Pétermann Y.-F.S.: On an estimate of Walfisz and Saltykov for an error term related to the Euler function. J. Théor. Nombr. Bordeaux 10, 203–236 (1998)zbMATHCrossRefGoogle Scholar
  7. 7.
    Saltykov, A.I.: On Euler’s function. [In Russian. Vestnik Moskov. Univ. Ser. I Mat. Meh. 6, 34–50 (1960)]Google Scholar
  8. 8.
    Sheng T.K.: Lines determined by lattice points in R 2. Nanta Math. 10, 77–81 (1977)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. http://www.research.att.com/~njas/sequences/
  10. 10.
    Suryanarayana D.: On the average order of the function \({E(x)=\sum_{n\le x} \phi(n)-3x^2/\pi^2}\) II. J. Indian Math. Soc. (N.S.) 42, 195–197 (1978)MathSciNetGoogle Scholar
  11. 11.
    Suryanarayana D., Sitaramachandra Rao R.: On the average order of the function \({E(x)=\sum_{n\le x}\phi(n)-3x^2/\pi^2}\). Ark. Mat. 10, 99–106 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Walfisz, A.: Weylsche Exponentialsummen in der neueren Zahlentheorie. VEB Deutsch. Verl. Wiss., Berlin (1963)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Anne-Maria Ernvall-Hytönen
    • 1
  • Kaisa Matomäki
    • 2
  • Pentti Haukkanen
    • 3
  • Jorma K. Merikoski
    • 3
  1. 1.Institutionen för matematik, Kungliga Tekniska HögskolanStockholmSweden
  2. 2.Department of MathematicsUniversity of TurkuTurkuFinland
  3. 3.Department of Mathematics and StatisticsUniversity of TampereTampereFinland

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