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Lower bounds for the largest eigenvalue of the gcd matrix on {1, 2,..., n}

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Abstract

Consider the n×n matrix with (i, j)’th entry gcd (i, j). Its largest eigenvalue λ n and sum of entries s n satisfy λ n > s n /n. Because s n cannot be expressed algebraically as a function of n, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S.Hong, R.Loewy (2004). We also conjecture that λ n > 6π−2 nlogn for all n. If n is large enough, this follows from F.Balatoni (1969).

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References

  1. E. Altınışık, A. Keskin, M. Yıldız, M. Demirbüken: On a conjecture of Ilmonen, Haukkanen and Merikoski concerning the smallest eigenvalues of certain GCD related matrices. Linear Algebra Appl. 493 (2016), 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  2. F. Balatoni: On the eigenvalues of the matrix of the Smith determinant. Mat. Lapok 20 (1969), 397–403. (In Hungarian.)

    MathSciNet  MATH  Google Scholar 

  3. S. Beslin, S. Ligh: Greatest common divisor matrices. Linear Algebra Appl. 118 (1989), 69–76.

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Hong, R. Loewy: Asymptotic behavior of eigenvalues of greatest common divisor matrices. Glasg. Math. J. 46 (2004), 551–569.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. A. Horn, C. R. Johnson: Matrix Analysis. Cambridge University Press, Cambridge, 2013.

    MATH  Google Scholar 

  6. D. S. Mitrinović, J. Sándor, B. Crstici: Handbook of Number Theory. Mathematics and Its Applications 351, Kluwer Academic Publishers, Dordrecht, 1995.

    Google Scholar 

  7. H. J. S. Smith: On the value of a certain arithmetical determinant. Proc. L. M. S. 7 (1875), 208–213.

    Article  MATH  Google Scholar 

  8. L. Tóth: A survey of gcd-sum functions. J. Integer Seq. (electronic only) 13 (2010), Article ID 10.8. 1, 23 pages.

    MathSciNet  MATH  Google Scholar 

  9. A. M. Yaglom, I. M. Yaglom: Non-elementary Problems in an Elementary Exposition. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moskva, 1954. (In Russian.)

    MATH  Google Scholar 

  10. E. W. Weisstein: Faulhaber’s Formula. From Mathworld—A Wolfram Web Resource, http://mathworld.wolfram. com/FaulhabersFormula. html.

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  1. School of Information Sciences, University of Tampere, Kanslerinrinne 1, FI-33014, Tampere, Finland

    Jorma K. Merikoski

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  1. Jorma K. Merikoski
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Correspondence to Jorma K. Merikoski.

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Dedicated to the memory of Miroslav Fiedler

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Merikoski, J.K. Lower bounds for the largest eigenvalue of the gcd matrix on {1, 2,..., n}. Czech Math J 66, 1027–1038 (2016). https://doi.org/10.1007/s10587-016-0307-5

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  • DOI: https://doi.org/10.1007/s10587-016-0307-5

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