Herglotz–Zagier function

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

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

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 
• 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 
• Zagier, Don (1975), "A Kronecker limit formula for real quadratic fields", Mathematische Annalen 213 (2): 153–184, doi:10.1007/BF01343950, ISSN 0025-5831