Selberg and refinement type inequalities on semi-Hilbertian spaces

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Sciences and Techniques, S. M. Ben Abdellah University, Fez, Morocco

2 Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco

3 Department of Mathematics, Faculty of Sciences, Ibn Tofai University, Kenitra, Morocco

Abstract

In this paper, we will study a type and refinement of Selberg type inequalities on semi-Hilbertian spaces, which is a simultaneous extension of the Bombieri type inequality in a semi-Hilbertian space. As applications, we give some examples of the Selberg inequality and its refinement on semi-Hilbertian spaces.

Keywords

[1] M.L. Arias, G. Corach and M.C. Gonzalez, Metric properties of projections in semi Hilbertian spaces, Integral
Equations and Operators Theory 62 (2008), 11–28.
[2] E. Bombieri, A note on the large sieve, Acta Arith. 18 (1971), 401–404.[3] H.G. Diamond, Elementary methods in the study of the distribution of prime numbers, Bull. Amer. Math. Soc. 7
(1982), 553–589.
[4] S.S. Dragomir, On the Boas-Belman in inner product spaces, arXiv:math/0307132v1 [math.CA] 9 Jul 2003
Aletheia University.
[5] P. Erd¨os, On a new method in elementary number theory which leads to an elementary proof of the prime number
theorem, Proc. Nat. Acad. Scis. USA. 35 (1949), 374–384.
[6] M. Fujii, Selberg inequality, ”http://www.kurims.kyoto-u.ac.jp/∼ kyodo/ kokyuroku /contents/ pdf/0743-
07.pdf”,(1991), 70–76.
[7] M. Fujii and R. Nakamoto, Simultaneous Extensions of Selberg inequality and Heinz-Kato-Furuta inequality,
Nihonkai Math. J. 9 (1998), 219–225.
[8] T. Furuta, When does the equality of a generalized Selbery inequality hold ?, Nihonkai Math. J. 2 (1991), 25–29.
[9] J. Hadamard, Sur la distribution des z´eros de la fonction zeta et ses cons´equences arithm´etiques, Bull. Soc. Math.
France 24 (1896), 199–220.
[10] H. Heilbronn, On the averages of some arithmetical functions of two variables, Mathematica 5 (1958), 1-7.
[11] J.E. Peˇcari´c, On some classical inequalities in unitary spaces, Mat. Bilten (Scopje) 16 (1992), 63–72.
[12] A. Selberg, An elementary proof of the prime number theorem, Ann. Math. 50 (1949), 305–313.
[13] J.J. Sylvester, On Tchebychef theorem of the totality of prime numbers comprised within given limits, Amer. J.
Math. 4 (1881), 230–247.
Volume 13, Issue 2
July 2022
Pages 1201-1206
  • Receive Date: 27 December 2020
  • Revise Date: 31 December 2020
  • Accept Date: 12 April 2021