Herglotz–Zagier function

In mathematics, the Herglotz–Zagier function, named after Gustav Herglotz and Don Zagier, is the function

${\displaystyle F(x)=\sum _{n=1}^{\infty }\left\{{\frac {\Gamma ^{\prime }(nx)}{\Gamma (nx)}}-\log(nx)\right\}{\frac {1}{n}}.}$

introduced by Zagier (1975) who used it to obtain a Kronecker limit formula for real quadratic fields.

References

• Herglotz, G. (1923), Berichte über die Verhandlungen der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, 75: 3–14 Missing or empty |title= (help)
• 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
• Zagier, Don (1975), "A Kronecker limit formula for real quadratic fields", Mathematische Annalen, 213: 153–184, doi:10.1007/BF01343950, ISSN 0025-5831, MR 0366877