# What is the compositional inverse of Riemann zeta function near $s=0$?

This question is related to my question here, I have used The Riemann-Siegle theta function particulary its taylor series arround $$t=0$$ to check wether Riemann zeta function has an explicite inversion formula near $$s=0$$ may it exist as a formal power series using lagrange inversion theorem since it is converges for $$|t|< \frac12$$ then :

What is the compositional inverse of Riemann zeta function near $$s=0$$ if it exists?

Note: The motivation of this question is to compute $$\zeta^{-1}(\dfrac{-1}{2})$$

Let $$U$$ be the image by $$\zeta$$ of $$|s| < 1/2$$ and for $$z \in U$$ $$F(z) = \frac{1}{2i\pi} \int_{|s| = 1/2} \frac{\zeta'(s) s}{\zeta(s)-z}ds$$

Then $$F$$ is analytic on $$U$$ and $$F(\zeta(s)) = s$$.

Proof : for $$z\in U, \exists |a| < 1/2, z = \zeta(a)$$ then for $$|s| \le 1/2$$, $$s \mapsto \frac{\zeta'(s) s}{z-\zeta(s)}$$ has a unique simple pole at $$s=a$$ of residue $$a$$ so that $$F(z) = Res(\frac{\zeta'(s) s}{\zeta(s)-z},a)= a$$.

Since $$\zeta(0) = -1/2$$ then $$F(-1/2) = 0$$.

When extending this local branch $$F$$ of $$\zeta^{-1}$$ by analytic continuation it will have a branch point at each $$z= \zeta(a), \zeta'(a)=0$$.

$$\zeta$$ isn't injective and for any $$a\ne 0$$ there are infinitely many $$s$$ such that $$\zeta(s)=a$$, infinitely many of them are on the strip $$\Re(s) \in (\sigma-\epsilon,\sigma+\epsilon)$$ for any $$\sigma > 1$$ such that $$1/\zeta(\sigma) < |a| < \zeta(\sigma)$$.

• Commented Aug 11, 2019 at 9:33