Herglotz–Zagier function: Difference between revisions
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In mathematics, the ''' |
In mathematics, the '''Herglotz–Zagier function''',{{r|m}} named after [[Gustav Herglotz]] and [[Don Zagier]],{{r|h|z}} is the function |
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:<math>F(x)= \sum^{\infty}_{n=1} \left\{\frac{\Gamma^{\prime}(nx)}{\Gamma (nx)} -\log (nx)\right\} \frac{1}{n}</math> |
:<math>F(x)= \sum^{\infty}_{n=1} \left\{\frac{\Gamma^{\prime}(nx)}{\Gamma (nx)} -\log (nx)\right\} \frac{1}{n}.</math> |
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==References== |
==References== |
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{{reflist|refs= |
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*{{citation|first=G. |last=Herglotz |journal=Ber. Verh. Sächs. Gesellschaft. Wiss. Leipzig Math.-Phys. Kl.|volume= 75 |year=1923|pages= 3--14}} |
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<ref name=h>{{citation|first=G. |last=Herglotz | authorlink=Gustav Herglotz| journal=Berichte über die Verhandlungen der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse<!-- "Reports on the Proceedings of the Royal Saxon Association of Sciences at Leipzig, Mathematical-Physical Classification" -->|title= |
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Über die Kroneckersche Grenzformel für reelle, quadratische Körper |volume= 75 |year=1923|pages= 3–14|jfm=49.0125.03}}</ref> |
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| ⚫ | <ref name=m>{{Citation | last1=Masri | first1=Riad | title=The Herglotz–Zagier function, double zeta functions, and values of L-series | doi=10.1016/j.jnt.2004.01.004 | mr=2059072 | year=2004 | journal=[[Journal of Number Theory]] | issn=0022-314X | volume=106 | issue=2 | pages=219–237| doi-access=free }}</ref> |
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| ⚫ | <ref name=z>{{Citation | last1=Zagier | first1=Don | authorlink=Don Zagier| title=A Kronecker limit formula for real quadratic fields | doi=10.1007/BF01343950 | mr=0366877 | year=1975 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=213 | issue=2 | pages=153–184| s2cid=54539768 }}</ref> |
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}} |
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{{DEFAULTSORT:Herglotz-Zagier function}} |
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[[Category: |
[[Category:Special functions]] |
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Latest revision as of 02:16, 24 March 2024
In mathematics, the Herglotz–Zagier function,[1] named after Gustav Herglotz and Don Zagier,[2][3] is the function
introduced by Zagier (1975) who used it to obtain a Kronecker limit formula for real quadratic fields.[3]
References
[edit]- ^ Masri, Riad (2004), "The Herglotz–Zagier function, double zeta functions, and values of L-series", Journal of Number Theory, 106 (2): 219–237, doi:10.1016/j.jnt.2004.01.004, ISSN 0022-314X, MR 2059072
- ^ Herglotz, G. (1923), "Über die Kroneckersche Grenzformel für reelle, quadratische Körper", Berichte über die Verhandlungen der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, 75: 3–14, JFM 49.0125.03
- ^ a b Zagier, Don (1975), "A Kronecker limit formula for real quadratic fields", Mathematische Annalen, 213 (2): 153–184, doi:10.1007/BF01343950, ISSN 0025-5831, MR 0366877, S2CID 54539768