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On Yoshida’s conjecture on the derivative of Shintani zeta functions. (English) Zbl 1225.11115

Summary: The purpose of this paper is to prove a conjecture in H. Yoshida’s book [Absolute CM-periods, Mathematical Surveys and Monographs 106. Providence, RI: AMS (2003; Zbl 1041.11001), p. 33] on the higher derivative of Shintani zeta functions at \(s=0\). We use multivariable Shintani zeta functions to prove the conjecture.

MSC:

11M41 Other Dirichlet series and zeta functions
11R42 Zeta functions and \(L\)-functions of number fields

Citations:

Zbl 1041.11001
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References:

[1] H. Hida, Elementary theory of \(L\)-functions and Eisenstein series , London Mathematical Society Student Texts, 26, Cambridge Univ. Press, Cambridge, 1993. · Zbl 0942.11024
[2] H. Yoshida, Absolute CM-periods , Mathematical Surveys and Monographs, 106, Amer. Math. Soc., Providence, RI, 2003.
[3] T. Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 2, 393-417. · Zbl 0349.12007
[4] T. Shintani, On a Kronecker limit formula for real quadratic fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 167-199. · Zbl 0364.12012
[5] T. Shintani, On values at \(s=1\) of certain \(L\) functions of totally real algebraic number fields, in Algebraic number theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976) , 201-212, Japan Soc. Promotion Sci., Tokyo. · Zbl 0363.12013
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