Series Expansion of a Cotangent Sum Related to the Estermann Zeta Function

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Authors: M. GOUBI

DOI: 10.46793/KgJMat2103.343G


In this paper, we study the cotangent sum c0(  )
  p related to the Estermann zeta function for the special case when the numerator is equal to 1 and get two useful series expansions of c0( 1 )


Estermann zeta function, Vasyunin cotangent sum, generating function.


[1]   L. Báez Duarte, M. Balazard, M. Landreau and E. Saias, Etude de l’autocorrlation multiplicative de la fonction partie fractionnaire, Ramanujan J. 9 (2005), 215–240.

[2]   S. Bettin and J. B. Conrey, Period functions and cotangent sums, Algebra and Number Theory 7(1) (2013), 215–242.

[3]   S. Bettin, On the distribution of a cotangent sum, Int. Math. Res. Not. IMRN 2015(21) (2015), 11419–11432.

[4]   G. B. Djordjević and G.  V.  Milovanović, Special Classes of Polynomials, University of Niš, Faculty of Technology, Leskovac, 2014.

[5]   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.

[6]   M. Goubi, Successive derivatives of Fibonacci type polynomials of higher order in two variables, Filomat 32(4) (2018), 5149—5159.

[7]   A. Bayad and M. Goubi, Reciprocity formulae for generalized Dedekind-Vasyunin-cotangent sums, Math. Methods Appl. Sci. 42(4) (2019), 1082–1098.

[8]   H. Maier, M. Th. Rassias, Explicit estimates of sums related to the Nyman-Beurling criterion for the Riemann hypothesis, J. Funct. Anal. DOI 10.1016/j.jfa.2018.06.022.

[9]   H. Maier, M. Th. Rassias, Generalizations of a cotangent sum associated to the Estermann zeta function, Commun. Contemp. Math. 18(1) (2016), DOI 10.1142/S0219199715500789.

[10]   M. Th. Rassias, A cotangent sum related to zeros of the Estermann zeta function, Appl. Math. Comput. 240 (2014), 161–167.

[11]   V. I. Vasyunin, On a biorthogonal system associated with the Riemann hypothesis, Algebra i Analiz 7(3) (1995), 118–135.