×

Real zeros of Dedekind zeta functions. (English) Zbl 1322.11118

The author applies some ideas of H. Kadiri [Int. J. Number Theory 8, No. 1, 125–147 (2012; Zbl 1279.11110)] and S. B. Stechkin [Math. Notes 8, 706–711 (1971); translation from Mat. Zametki 8, 419–429 (1970; Zbl 0233.10020)] to enlarge the value of \(c\) for which the interval \((1-c/\log d(K),1)\) contains at most \(m\) roots of the Dedekind zeta-function \(\zeta_K(s)\).
In [Manuscr. Math. 125, No. 1, 43–67 (2008; Zbl 1137.11072), Corollary 9], the author showed for non-normal sextic \(CM\)-fields \(K\), not containing any imaginary quadratic subfield, and whose totally real cubic subfield \(F\) is non-normal, that if \(h^-(K)=1\) and \(2\) splits in \(F\), then \(d(K)\leq 50\cdot10^{24}\). The new value of \(c\) permits to diminish this bound to \(d(K)\leq 4\cdot10^{24}\).

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11R21 Other number fields
11R29 Class numbers, class groups, discriminants
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1016/j.jnt.2012.02.020 · Zbl 1275.11143 · doi:10.1016/j.jnt.2012.02.020
[2] Boutteaux G., Acta Math. Inform. Univ. Ostraviensis 10 pp 3– (2002)
[3] DOI: 10.4153/CJM-1973-090-3 · Zbl 0272.12010 · doi:10.4153/CJM-1973-090-3
[4] Hoffstein J., Acta Arith. 38 pp 167– (1980)
[5] DOI: 10.1142/S1793042112500078 · Zbl 1279.11110 · doi:10.1142/S1793042112500078
[6] DOI: 10.5802/jtnb.257 · Zbl 1010.11063 · doi:10.5802/jtnb.257
[7] DOI: 10.1090/S0002-9947-03-03313-0 · Zbl 1026.11085 · doi:10.1090/S0002-9947-03-03313-0
[8] DOI: 10.1112/S0024610705006654 · Zbl 1078.11064 · doi:10.1112/S0024610705006654
[9] DOI: 10.4064/aa121-3-1 · Zbl 1122.11053 · doi:10.4064/aa121-3-1
[10] DOI: 10.1007/s00229-007-0132-0 · Zbl 1137.11072 · doi:10.1007/s00229-007-0132-0
[11] Pintz J., Acta Arith. 32 pp 163– (1977)
[12] DOI: 10.1007/BF01405166 · Zbl 0278.12005 · doi:10.1007/BF01405166
[13] DOI: 10.1007/BF01104369 · Zbl 0233.10020 · doi:10.1007/BF01104369
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.