Let, $a \in \Bbb R$ be any real number with $a>1$. Define, $f:\Bbb R\to \Bbb R$ by $$f(x)=-\sum_{n=1}^\infty\frac{\sin (x\log n)}{n^a},\forall x\in \Bbb R.$$ I have managed to find a way to show it has a fixed point in $\Bbb R$ by Brower's fixed point theorem [which is obviously zero].
I want to show it is unique. I am unable to proceed with the known fixed point theorems. After some regular homework on that series, I have concluded that if we are able to show that, "$f(x)$ attains its upper bound/maximum after $x=\zeta(a)$ where $\zeta$- is the Riemann Zeta function then the fixed point will be unique [Not verified yet!]."
Loosely speaking, the problem is directly related to the fact that the function $f_a(x)=\operatorname{Im}(\zeta(a+ix)),x\in\Bbb R$ has a unique fixed point [precisely zero] for every $a>1$.
How should I proceed? Any related paper/note/answer is highly appreciated.
Thanks in advance!
Is $(1,\infty)$ the whole vanishing set of $\Im(\zeta(s) - s), \Re(s) > 1$ ?
For $\Re(s)$ large enough it follows from that $|\zeta'(s)| < 1$.
For $\Re(s) = 1+\epsilon$ use that $\zeta(s) = \frac{1}{s-1}+O(\log (2+|\Im(s)|))$ uniformly on $\Re(s) > 1$ which restricts the possible zeros of $\Im(\zeta(s) - s)$ to a bounded domain where you can do the numerical checks