# An equation with Gamma Euler function in critical strip

Let $$D=\{z \in \mathbb{C} : 0<\Re(z)<\frac{1}{2} \text{ or } \frac{1}{2}<\Re(z)<1 \}$$ that is the critical strip without critical line.

I have to find if the following equation, with Gamma Euler function, has any root in $$D$$ $$\Gamma(z) = \dfrac {\pi^z} {\cos \left( \dfrac{\pi}{2} \cdot z \right) \cdot 2^{1-z} } \cdot \dfrac {1-2^{1-z}} {1-2^z}$$

Thanks.

Imprecise answer: if you denote by $$f(z)$$ the log of the ratio of the two sides, we have $$f(1-z)=-f(z)$$ (I assume you constructed your function in that way). One now uses an old theorem of Hermite, unfortunately I don't remember the exact statement and reference (he states it for polynomials, but it is easily generalized) which shows that all the zeros have real part $$\ge1/2$$, hence by the functional equation also $$\le1/2$$, so are on the critical strip. Sorry to be so imprecise (same proof applies to functions such as $$\Lambda(s+a)+\Lambda(s-a)$$ with $$a\ge1/2$$ for instance, with $$\Lambda(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$$).
• I'm interested to proof if my equation has got any root in $D$. By your answer I don't understand if there are surely roots in $D$ or it's just a chance. – Matey Math Mar 4 at 22:11
• If my observation of the use of Hermite's theorem is correct, this implies that there are NO roots in $D$ since you exclude the critical line. – Henri Cohen Mar 5 at 9:20