Modulus of Riemann zeta function \zeta(\sigma+it) as a real function of t

Question

Define

$$ Y(t) := |\zeta(\sigma_0 + it)|, \ \ \ \ \ \ (\frac{1}{2} < \sigma_0 < 1)$$

Is $Y(t)$ an analytic function of the real variable $t$?

Remark Hardy defined the function $$ Z(t) := \zeta(\frac{1}{2} + it) (\chi(\frac{1}{2} + it))^{-1/2},$$ and showed that $$ |Z(t)| = |\zeta(\frac{1}{2} + it)|.$$ Of course, $$ \zeta(\frac{1}{2} + it) (\chi(\frac{1}{2} + it))^{-1/2}$$ is analytic in a neighborhood of $(1/2,t)$ in the complex plane, and therefore real analytic in $t$.

Answer 1

For $f$ meromorphic then $(x,y)\to |f(x+iy)|$ is real-analytic away from the poles and the zeros-of-odd-order of $f$.

Proof : at the poles of $f$ the non-analyticity is obvious,

away from the poles $(x,y)\to \Re(f(x+iy))$ is real-analytic thus so is $(x,y)\to |f(x+iy)|^2$,

$u\to u^{1/2}$ is analytic on $(0,\infty)$,

at a zero of even order then $f=g^2$ with $g$ analytic,

it remains to check that $|f(x+iy)|$ is not smooth at the zeros of odd order, which follows from $f(z) = (z-\rho)^{2k+1} g(z),g(\rho)\ne 0$.

Note from Conard's answer : near an odd order zero $f(x_0+iy_0)=0$ the function $y\to sign(y-y_0) |f(x_0+iy)|$ is analytic, thus on OP's vertical line $\sigma_0+i\Bbb{R}$ the real analyticity can be repaired through multiplication by some $(-1)^{h_{\sigma_0}(y)}$ term, where $h_{\sigma_0}$ is counting the number of (odd order) zeros on the line.

Answer 2

The OP is utterly wrong as $Z(t)$ is a real analytic function defined as

$Z(t) = \zeta(\frac{1}{2} + it) (\chi(\frac{1}{2} + it))^{-1/2}$ which satisfies:

$|Z(t)|=|\zeta(\frac{1}{2} + it)|$ but it is simply not true that $Z(t)=|\zeta(\frac{1}{2} + it)|$

What is true is that:

$Z(t) = \pm |\zeta(\frac{1}{2} + it)|$ and a recent observation of Ivic shows that if $t$ is not an ordinate of a zero of $\zeta$ (not necessarily a critical zero btw, could be a presumable counterexample to RH zero), then

$Z(t)=(-1)^{N(t)+1}|\zeta(\frac{1}{2} + it)|, t>0$, where $N(t)=\sum_{0 < \gamma \le t}1$ the number of (non-trivial but could be outside critical line if any such) zeroes of RZ up to and including imaginary level $t$ (which is the definition in Titchmarsh standard textbook and not the usual convention of taking arithmetic functions with jumps to be adding half the value at jumps - there is more discussion of this subtle distinction in the note linked above)