I am self studying analytic number theory from Tom M Apostol Modular functions and Dirichlet series in number theory and I am having a doubt in Theorem 3.1 . ( I have doubt only in highlighted part of image 3)
adding Image of theorem
I have doubt only in highlighted part of theorem in image 3 . I am not able to understand how Apostol writes that the function $zF_n(z) $ has as n->$\infty$ , the limit 1/8 on the edges connecting y, i, ?
Can someone please tell how to deduce it.
Best Answer
I think that the question is a simple one. The function $\,\exp(n z)\to 0\,$ as $\,n\to \infty\,$ iff the real part of $\,z\,$ is negative. This implies that $\,\cot(nz)\to i\,$ as $\,n\to \infty\,$ if the real part of $\,z\,$ is negative, but $\,-i\,$ if the real part of $\,z\,$ is positive. This implies that the first factor of cotangent in $\,zF_n(z)\,$ converges to $\,i\,$ if the real part of $\,z\,$ is positive and to $\,-i\,$ if the real part of $\,z\,$ is negative. The second factor of cotangent converges to $\,i\,$ if the imaginary part of $\,z\,$ is negative and to $\,-i\,$ if the imaginary part of $\,z\,$ is positive.
Finally $\,zF_n(z)\to \frac18\,$ or $\,-\frac18\,$ according to the quadrant that $\,z\,$ is in.