The growth of Dirichlet series

Zhendong Gu; Daochun Sun

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 1, page 29-38
  • ISSN: 0011-4642

Abstract

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We define Knopp-Kojima maximum modulus and the Knopp-Kojima maximum term of Dirichlet series on the right half plane by the method of Knopp-Kojima, and discuss the relation between them. Then we discuss the relation between the Knopp-Kojima coefficients of Dirichlet series and its Knopp-Kojima order defined by Knopp-Kojima maximum modulus. Finally, using the above results, we obtain a relation between the coefficients of the Dirichlet series and its Ritt order. This improves one of Yu Jia-Rong's results, published in Acta Mathematica Sinica 21 (1978), 97–118. We also give two examples to show that the condition under which the main result holds can not be weakened.

How to cite

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Gu, Zhendong, and Sun, Daochun. "The growth of Dirichlet series." Czechoslovak Mathematical Journal 62.1 (2012): 29-38. <http://eudml.org/doc/246707>.

@article{Gu2012,
abstract = {We define Knopp-Kojima maximum modulus and the Knopp-Kojima maximum term of Dirichlet series on the right half plane by the method of Knopp-Kojima, and discuss the relation between them. Then we discuss the relation between the Knopp-Kojima coefficients of Dirichlet series and its Knopp-Kojima order defined by Knopp-Kojima maximum modulus. Finally, using the above results, we obtain a relation between the coefficients of the Dirichlet series and its Ritt order. This improves one of Yu Jia-Rong's results, published in Acta Mathematica Sinica 21 (1978), 97–118. We also give two examples to show that the condition under which the main result holds can not be weakened.},
author = {Gu, Zhendong, Sun, Daochun},
journal = {Czechoslovak Mathematical Journal},
keywords = {Dirichlet series; order; abscissa of convergence; Dirichlet series; abscissa of convergence},
language = {eng},
number = {1},
pages = {29-38},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The growth of Dirichlet series},
url = {http://eudml.org/doc/246707},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Gu, Zhendong
AU - Sun, Daochun
TI - The growth of Dirichlet series
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 1
SP - 29
EP - 38
AB - We define Knopp-Kojima maximum modulus and the Knopp-Kojima maximum term of Dirichlet series on the right half plane by the method of Knopp-Kojima, and discuss the relation between them. Then we discuss the relation between the Knopp-Kojima coefficients of Dirichlet series and its Knopp-Kojima order defined by Knopp-Kojima maximum modulus. Finally, using the above results, we obtain a relation between the coefficients of the Dirichlet series and its Ritt order. This improves one of Yu Jia-Rong's results, published in Acta Mathematica Sinica 21 (1978), 97–118. We also give two examples to show that the condition under which the main result holds can not be weakened.
LA - eng
KW - Dirichlet series; order; abscissa of convergence; Dirichlet series; abscissa of convergence
UR - http://eudml.org/doc/246707
ER -

References

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  9. Valiron, G., Sur la croissance du module maximum des séries entières, S. M. F. Bull. 44 (1916), 45-64 French. (1916) MR1504749
  10. Valiron, G., Sur l'abscisse de convergence des séries de Dirichlet, S. M. F. Bull. 52 (1924), 166-174 French. (1924) MR1504844
  11. Valiron, G., 10.1073/pnas.20.3.211, Proc. Natl. Acad. Sci. USA 20 (1934), 211-215. (1934) Zbl0009.02503DOI10.1073/pnas.20.3.211
  12. Valiron, G., Théorie Générale des Séries de Dirichlet, Paris, Gauthier-Villars (Mémorial des sciences mathématiques, fasc. 17) (1926), French pp. 56. (1926) 
  13. Yu, Ch.-Y., Dirichlet series, Analytic functions of one complex variable, Contemp. Math., 48, Amer. Math. Soc., Providence, RI (1985), 201-216. (1985) MR0838106
  14. Yu, Ch.-Y., 10.24033/asens.986, Ann. Sci. Éc. Norm. Supér, III. Sér. 68 (1951), 65-104 French. (1951) Zbl0045.03802MR0041223DOI10.24033/asens.986
  15. Yu, J.-R., Some properties of random Dirichlet series, Acta Math. Sin. 21 (1978), 97-118 Chinese. (1978) Zbl0386.60044MR0507192

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