Regarding Fourier coefficients of entire modular forms and

Question

I am self studying Analytic number theory from Tom M Apostol and I cannot think about a deduction in proof of a theorem in Chapter 6 ( Modular forms with multiplicative coefficients) .

In section 6.15 - Estimates for the Fourier coefficients of Entire forms

Apostol assumes f to be an entire form with Fourier expansion

f($\tau$) = $\sum_{n=0}^\infty = c(n) x^n $ . ----(1)

Then later in same page Apostol mentions

The series in (1) converges absolutely if |x| <1 .

I am unable to think why f($\tau$) converges absolutely for |x|<1 although there is no restriction on coefficients c(n) ? Why does value of coefficients c(n) doesn't matters and what theorem or result is Apostol actually using to deduce this.

My second doubt is -- Apostol later writes If $\tau $ is in fundamental region of $\Gamma$ then $\tau$ = u +iv with v $\ge$ $\frac {√3} {2} $

Can somebody please tell why v couldn't be less than $\frac {√3} {2} $ ?

Answer 1

The word "entire form" doesn't have any meaning. If $f(z)$ is analytic and $1$-periodic on $\Im(z) > 0$ then $F(q)=f(\frac{\log q}{2i\pi }) $ is analytic on $0<|q|< 1$, if $f$ is bounded on $\Im(z) > 1$ then $q^2 F(q)$ is holomorphic thus analytic on $|q|<1$ thus (Cauchy integral formula) it is equal to its Taylor series $F(q)=\sum_{n\ge 0} c(n)q^n$ which converges, thus absolutely, for $|q|<1$ and $f(z)=\sum_{n\ge 0} c(n) e^{2i\pi nz}$. Without the boundedness assumption we have the Laurent series $F(q)=\sum_{n\in \Bbb{Z}} c(n)q^n$ which converges absolutely for $0<|q|< 1$ and $f(z)=\sum_{n\in \Bbb{Z}} c(n) e^{2i\pi nz}$.