On the error term of the logarithm of the lcm of a quadratic sequence

Juanjo Rué[1]; Paulius Šarka[2]; Ana Zumalacárregui[3]

  • [1] Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) Nicolás Cabrera 13-15 28049 Madrid, Spain
  • [2] Institute of Mathematics and Informatics Akademijos 4 08663 Vilnius, Lithuania and Department of Mathematics and Informatics, Vilnius University Naugarduko 24 03225 Vilnius, Lithuania
  • [3] Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departmento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid, Spain

Journal de Théorie des Nombres de Bordeaux (2013)

  • Volume: 25, Issue: 2, page 457-470
  • ISSN: 1246-7405

Abstract

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We study the logarithm of the least common multiple of the sequence of integers given by 1 2 + 1 , 2 2 + 1 , ⋯ , n 2 + 1 . Using a result of Homma [5] on the distribution of roots of quadratic polynomials modulo primes we calculate the error term for the asymptotics obtained by Cilleruelo [3].

How to cite

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Rué, Juanjo, Šarka, Paulius, and Zumalacárregui, Ana. "On the error term of the logarithm of the lcm of a quadratic sequence." Journal de Théorie des Nombres de Bordeaux 25.2 (2013): 457-470. <http://eudml.org/doc/275750>.

@article{Rué2013,
abstract = {We study the logarithm of the least common multiple of the sequence of integers given by $1^2+1, 2^2+1,\dots , n^2+1$. Using a result of Homma [5] on the distribution of roots of quadratic polynomials modulo primes we calculate the error term for the asymptotics obtained by Cilleruelo [3].},
affiliation = {Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) Nicolás Cabrera 13-15 28049 Madrid, Spain; Institute of Mathematics and Informatics Akademijos 4 08663 Vilnius, Lithuania and Department of Mathematics and Informatics, Vilnius University Naugarduko 24 03225 Vilnius, Lithuania; Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departmento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid, Spain},
author = {Rué, Juanjo, Šarka, Paulius, Zumalacárregui, Ana},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {quadratic sequence; quadratic polynomial; lcm; Vinogradov symbol},
language = {eng},
month = {9},
number = {2},
pages = {457-470},
publisher = {Société Arithmétique de Bordeaux},
title = {On the error term of the logarithm of the lcm of a quadratic sequence},
url = {http://eudml.org/doc/275750},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Rué, Juanjo
AU - Šarka, Paulius
AU - Zumalacárregui, Ana
TI - On the error term of the logarithm of the lcm of a quadratic sequence
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/9//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 2
SP - 457
EP - 470
AB - We study the logarithm of the least common multiple of the sequence of integers given by $1^2+1, 2^2+1,\dots , n^2+1$. Using a result of Homma [5] on the distribution of roots of quadratic polynomials modulo primes we calculate the error term for the asymptotics obtained by Cilleruelo [3].
LA - eng
KW - quadratic sequence; quadratic polynomial; lcm; Vinogradov symbol
UR - http://eudml.org/doc/275750
ER -

References

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  1. P. Bateman, J. Kalb and A. Stenger, A limit involving least common multiples. Amer. Math. Monthly 109 (2002), 393–394. Zbl1124.11300
  2. P. L. Chebishev, Memoire sur les nombres premiers. J. de Math. Pures et Appl. 17 (1852), 366–390. 
  3. J. Cilleruelo, The least common multiple of a quadratic sequence. Compos. Math. 147 (2011), no.4, 1129–1150. Zbl1248.11068MR2822864
  4. W. Duke, J. Friedlander and H. Iwaniec, Equidistribution of roots of a quadratic congruence to prime moduli. Ann. of Math. 141 (1995), no.2, 423–441. Zbl0840.11003MR1324141
  5. K. Homma, On the discrepancy of uniformly distributed roots of quadratic congruences. J. of Number Theory 128 (2008), no.3, 500–508. Zbl1195.11089MR2389853
  6. S. Hong, G. Quian and Q. Tan, The least common multiple of sequence of product of linear polynomials. Acta Math. Hungar. 135 (2012), no.1-2, 160-167. Zbl1265.11093MR2898796
  7. H. L. Montgomery and R. C. Vaughan, Multiplicative number theory. I. Classical theory. Cambridge University Press, 2007. Zbl1245.11002MR2378655
  8. P. Moree, Counting Numbers in multiplicative sets: Landau versus Ramanujan. Šiauliai Math. Semin., to appear. Zbl1315.11083MR3012680
  9. H. Niederreiter, Random number generation and quasi-Monte Carlo methods. CBMS-NSF Regional Conference Series in Applied Mathematics 63 , Society for Industrial and Applied Mathematics (SIAM), 1992. Zbl0761.65002MR1172997
  10. Á. Tóth, Roots of quadratic congruences. Internat. Math. Res. Notices 14 (2000), 719–739. Zbl1134.11339MR1776618

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