The Riemann hypothesis and functional equations for zeta functions over \(\mathbb F_1\). (English) Zbl 1275.11125

Kim, Sojung; Koyama, Shin-ya; Kurokawa, Nobushige

Proc. Japan Acad., Ser. A 85, No. 6, 75-80 (2009).

Summary: We prove functional equations for the absolute zeta functions. We also show that the absolute zeta functions satisfy the tensor structure in the sense that their singularities possess an additive property under the tensor product. Moreover those singularities satisfy the analog of the Riemann hypothesis.

MSC:

11M38	Zeta and \(L\)-functions in characteristic \(p\)
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

Keywords:

zeta functions; field with one element; absolute mathematics

References:

[1]	A. Deitmar, S. Koyama and N. Kurokawa, Absolute zeta functions, Proc. Japan Acad. Ser. A Math. Sci. 84A (2008), no. 8, 138-142. · Zbl 1225.11113 · doi:10.3792/pjaa.84.138
[2]	S. Koyama and N. Kurokawa, Multiple zeta functions: the double sine function and the signed double Poisson summation formula, Compos. Math. 140 (2004), no. 5, 1176-1190. · Zbl 1135.11329 · doi:10.1112/S0010437X04000521
[3]	S. Koyama and N. Kurokawa, Multiple Euler products , Proceedings of the St. Petersburg Mathematical Society 11 (2005) 123-166, (in Russian). American Mathematical Society Translation, Series 2, 218 (2006) 101-140.
[4]	N. Kurokawa, Zeta functions over \(\F_ 1\), Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 10, 180-184. · Zbl 1141.11316 · doi:10.3792/pjaa.81.180
[5]	N. Kurokawa, Weighted Euler products. (preprint). · Zbl 1275.92079
[6]	Y. Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa), Astérisque 228 (1995), 121-163. · Zbl 0840.14001