Shiomi, Daisuke Determinant formulas for zeta functions for real abelian function fields. (English) Zbl 1286.11190 Proc. Japan Acad., Ser. A 87, No. 10, 183-185 (2011). There are many “determinant formulas” for class numbers of cyclotomic fields, the most famous perhaps being Maillet’s determinant. M. Rosen [Proc. Am. Math. Soc. 125, No. 5, 1299–1303 (1997; Zbl 0872.11045)] proved a similar formula for relative class numbers of cyclotomic function fields. In [Acta Arith. 138, No. 3, 259–268 (2009; Zbl 1228.11138)], the author explained these formulas by deriving a “determinant formula” for the zeta function of such fields in the special case of maximal real subfields of cyclotomic function fields. In this article, this result is generalized to arbitrary real subfields of cyclotomic function fields. Reviewer: Franz Lemmermeyer (Jagstzell) MSC: 11R42 Zeta functions and \(L\)-functions of number fields 11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.) Keywords:cyclotomic function field; Maillet determinant; zeta function; Carlitz module Citations:Zbl 0872.11045; Zbl 1228.11138 PDFBibTeX XMLCite \textit{D. Shiomi}, Proc. Japan Acad., Ser. A 87, No. 10, 183--185 (2011; Zbl 1286.11190) Full Text: DOI References: [1] H. Jung, S. Bae and J. Ahn, Determinant formulas for class numbers in function fields, Math. Comp. 74 (2005), no. 250, 953-965. · Zbl 1082.11075 · doi:10.1090/S0025-5718-04-01671-0 [2] J. Ahn, S. Choi and H. Jung, Class number formulae in the form of a product of determinants in function fields, J. Aust. Math. Soc. 78 (2005), no. 2, 227-238. · Zbl 1092.11044 · doi:10.1017/S1446788700008053 [3] D. R. Hayes, Explicit class field theory for rational function fields, Trans. Amer. Math. Soc. 189 (1974), 77-91. · Zbl 0292.12018 · doi:10.2307/1996848 [4] M. Rosen, A note on the relative class number in function fields, Proc. Amer. Math. Soc. 125 (1997), no. 5, 1299-1303. · Zbl 0872.11045 · doi:10.1090/S0002-9939-97-03748-9 [5] M. Rosen, Number theory in function fields , Graduate Texts in Mathematics, 210, Springer, New York, 2002. · Zbl 1043.11079 [6] D. Shiomi, A determinant formula for congruence zeta functions of maximal real cyclotomic function fields, Acta Arith. 138 (2009), no. 3, 259-268. · Zbl 1228.11138 · doi:10.4064/aa138-3-3 [7] L. C. Washington, Introduction to cyclotomic fields , Graduate Texts in Mathematics, 83, Springer, New York, 1982. · Zbl 0484.12001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.