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On Bell polynomials associated to Vasyunin cotangent sums

Year 2022, Volume: 31 Issue: 31, 230 - 242, 17.01.2022
https://doi.org/10.24330/ieja.1058435

Abstract

The present work is focused on the study of a cotangent sum
associated to the zeros of the Estermann zeta function and Riemann
zeta function. We use Bell polynomials and generating functions
approach to give arithmetical proof of its Dirichlet series
different from that given by M. Th. Rassias.

References

  • T. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
  • L. Baez Duarte, M. Balazard, M. Landreau and E. Saias, Etude de lutocorrelation multiplicative de la fonction partie fractionnaire, Ramanujan J., 9 (2005), 215-240.
  • A. Bayad and M. Goubi, Reciprocity formulae for generalized Dedekind- Vasyunin-cotangent sums, Math. Methods Appl. Sci., 42(4) (2019), 1082-1098.
  • S. Belhadj and M. Goubi, On the Vasyunin Cotangent sums related to Riemann Hypothesis, WSEAS Transactions on Mathematics, 19 (2020), 676-682.
  • L. Comtet, Advanced Combinatorics, Reidel, Boston, 1974.
  • F. Caldarola, M. Maiolo and V. Solferino, A new approch of Z-transform through infinite computation, Commun. Nonlinear Sci. Numer. Simulat., 82 (2020), article id. 105019.
  • R. de la Breteche and G. Tenenbaum, Series trigonometriques a coeffcients arithmetiques, J. Anal. Math., 92 (2004), 1-79.
  • F. Faa di Bruno, Sullo Sviluppo delle funzioni, Annali di Scienze Matematichee fisiche di Tortolini, 6 (1855), 479-480.
  • T. Fukaya, Hasse Zeta Functions of Non-commutative Rings, J. Algebra, 208(1) (1998), 304-342.
  • M. Goubi, Series expansion of a cotangent sum related to the Estermann zeta function, Kragujevac J. Mathematics, 45(3) (2021), 343-352.
  • M. Goubi, On composition of generating functions, Casp. J. Math. Sci., 9(2) (2020), 256-265.
  • M. Goubi, Explicit formula of a new class of q-Hermite-based Apostol-type polynomials and generalization, Notes on Number Theory and Discrete Mathematics, 26(4) (2020), 93-102.
  • M. Goubi, A. Bayad and M. O. Hernane, Explicit and asymptotic formulae for Vasyunin-cotangent sums, Publ. Inst. Math. (Beograd) (N.S.), 102(116) (2017), 155-174.
  • F. J. Grunewald, D. Segal and G. C. Smith, Subgroups of finite index in nilpotent groups, Invent. Math., 93(1) (1988), 185-223.
  • M. Ishibashi, The value of the Estermann zeta function at s = 0, Acta Arith., 73(4) (1995), 357-361.
  • E. I. Jury, Theory and Application of Z-Transform Method, John Wiley & Sons, New York, 1964.
  • H. Maier and M. Th. Rassias, Generalizations of a cotangent sum associated to the Estermann zeta function, Commun. Contemp. Math., 18(1) (2016), 1550078 (89 pp).
  • C. Papachristodoulos and M. Papadimitrakis, On universality and convergence of the Fourier series of functions in the disc algebra, J. Anal. Math., 137(1) (2019), 57-71.
  • J. Ragazzini and L. Zadeh, The analysis of sampled-data systems, Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry, 71(5) (1952), 225-234.
  • M. Th. Rassias, Analytic Investigation of Cotangent Sums to the Riemann Zeta Function, Zurich, 2014.
  • M. Th. Rassias, A cotangent sum related to zeros of the Estermann zeta function, Appl. Math. Comput., 240 (2014), 161-167.
  • S. Roman, The Formula of Faa di Bruno, Amer. Math. Monthly, 87(10) (1980), 805-809.
  • L. Solomon, Zeta functions and integral representation theory, Advances in Math., 26(3) (1977), 306-326.
  • V. I. Vasyunin, On a biorthogonal system associated with the Riemann hypothesis, Algebra i Analiz, 7(3) (1995), 118-135.
  • C. Voll, Functional equations for zeta functions of groups and rings, Ann. of Math., 172(2) (2010), 1181-1218.
Year 2022, Volume: 31 Issue: 31, 230 - 242, 17.01.2022
https://doi.org/10.24330/ieja.1058435

Abstract

References

  • T. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
  • L. Baez Duarte, M. Balazard, M. Landreau and E. Saias, Etude de lutocorrelation multiplicative de la fonction partie fractionnaire, Ramanujan J., 9 (2005), 215-240.
  • A. Bayad and M. Goubi, Reciprocity formulae for generalized Dedekind- Vasyunin-cotangent sums, Math. Methods Appl. Sci., 42(4) (2019), 1082-1098.
  • S. Belhadj and M. Goubi, On the Vasyunin Cotangent sums related to Riemann Hypothesis, WSEAS Transactions on Mathematics, 19 (2020), 676-682.
  • L. Comtet, Advanced Combinatorics, Reidel, Boston, 1974.
  • F. Caldarola, M. Maiolo and V. Solferino, A new approch of Z-transform through infinite computation, Commun. Nonlinear Sci. Numer. Simulat., 82 (2020), article id. 105019.
  • R. de la Breteche and G. Tenenbaum, Series trigonometriques a coeffcients arithmetiques, J. Anal. Math., 92 (2004), 1-79.
  • F. Faa di Bruno, Sullo Sviluppo delle funzioni, Annali di Scienze Matematichee fisiche di Tortolini, 6 (1855), 479-480.
  • T. Fukaya, Hasse Zeta Functions of Non-commutative Rings, J. Algebra, 208(1) (1998), 304-342.
  • M. Goubi, Series expansion of a cotangent sum related to the Estermann zeta function, Kragujevac J. Mathematics, 45(3) (2021), 343-352.
  • M. Goubi, On composition of generating functions, Casp. J. Math. Sci., 9(2) (2020), 256-265.
  • M. Goubi, Explicit formula of a new class of q-Hermite-based Apostol-type polynomials and generalization, Notes on Number Theory and Discrete Mathematics, 26(4) (2020), 93-102.
  • M. Goubi, A. Bayad and M. O. Hernane, Explicit and asymptotic formulae for Vasyunin-cotangent sums, Publ. Inst. Math. (Beograd) (N.S.), 102(116) (2017), 155-174.
  • F. J. Grunewald, D. Segal and G. C. Smith, Subgroups of finite index in nilpotent groups, Invent. Math., 93(1) (1988), 185-223.
  • M. Ishibashi, The value of the Estermann zeta function at s = 0, Acta Arith., 73(4) (1995), 357-361.
  • E. I. Jury, Theory and Application of Z-Transform Method, John Wiley & Sons, New York, 1964.
  • H. Maier and M. Th. Rassias, Generalizations of a cotangent sum associated to the Estermann zeta function, Commun. Contemp. Math., 18(1) (2016), 1550078 (89 pp).
  • C. Papachristodoulos and M. Papadimitrakis, On universality and convergence of the Fourier series of functions in the disc algebra, J. Anal. Math., 137(1) (2019), 57-71.
  • J. Ragazzini and L. Zadeh, The analysis of sampled-data systems, Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry, 71(5) (1952), 225-234.
  • M. Th. Rassias, Analytic Investigation of Cotangent Sums to the Riemann Zeta Function, Zurich, 2014.
  • M. Th. Rassias, A cotangent sum related to zeros of the Estermann zeta function, Appl. Math. Comput., 240 (2014), 161-167.
  • S. Roman, The Formula of Faa di Bruno, Amer. Math. Monthly, 87(10) (1980), 805-809.
  • L. Solomon, Zeta functions and integral representation theory, Advances in Math., 26(3) (1977), 306-326.
  • V. I. Vasyunin, On a biorthogonal system associated with the Riemann hypothesis, Algebra i Analiz, 7(3) (1995), 118-135.
  • C. Voll, Functional equations for zeta functions of groups and rings, Ann. of Math., 172(2) (2010), 1181-1218.
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Samir Belhadj This is me

Mouloud Goubi This is me

Publication Date January 17, 2022
Published in Issue Year 2022 Volume: 31 Issue: 31

Cite

APA Belhadj, S., & Goubi, M. (2022). On Bell polynomials associated to Vasyunin cotangent sums. International Electronic Journal of Algebra, 31(31), 230-242. https://doi.org/10.24330/ieja.1058435
AMA Belhadj S, Goubi M. On Bell polynomials associated to Vasyunin cotangent sums. IEJA. January 2022;31(31):230-242. doi:10.24330/ieja.1058435
Chicago Belhadj, Samir, and Mouloud Goubi. “On Bell Polynomials Associated to Vasyunin Cotangent Sums”. International Electronic Journal of Algebra 31, no. 31 (January 2022): 230-42. https://doi.org/10.24330/ieja.1058435.
EndNote Belhadj S, Goubi M (January 1, 2022) On Bell polynomials associated to Vasyunin cotangent sums. International Electronic Journal of Algebra 31 31 230–242.
IEEE S. Belhadj and M. Goubi, “On Bell polynomials associated to Vasyunin cotangent sums”, IEJA, vol. 31, no. 31, pp. 230–242, 2022, doi: 10.24330/ieja.1058435.
ISNAD Belhadj, Samir - Goubi, Mouloud. “On Bell Polynomials Associated to Vasyunin Cotangent Sums”. International Electronic Journal of Algebra 31/31 (January 2022), 230-242. https://doi.org/10.24330/ieja.1058435.
JAMA Belhadj S, Goubi M. On Bell polynomials associated to Vasyunin cotangent sums. IEJA. 2022;31:230–242.
MLA Belhadj, Samir and Mouloud Goubi. “On Bell Polynomials Associated to Vasyunin Cotangent Sums”. International Electronic Journal of Algebra, vol. 31, no. 31, 2022, pp. 230-42, doi:10.24330/ieja.1058435.
Vancouver Belhadj S, Goubi M. On Bell polynomials associated to Vasyunin cotangent sums. IEJA. 2022;31(31):230-42.