A detailed note on the zeros of Eisenstein series for ${\Gamma }_0^{\ast }(5)$ and ${\Gamma }_0^{\ast }(7)$

Abstract

The present paper provides the details omitted from the more concise study "On the zeros of Eisenstein series for ${\Gamma }_0^{\ast }(5)$ and ${\Gamma }_0^{\ast }(7)$." We locate almost all of the zeros of the Eisenstein series associated with the Fricke groups of level 5 and 7 in their fundamental domains by applying and extending the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (1970). We also use the arguments of some terms of the Eisenstein series in order to improve existing error bounds.

arXiv comment39 pages, 3 Postscript figures. http://researchmap.jp/j_shigezumi/