I am missing some steps in the final derivation of a probabilistic computation of the even values of $\zeta$. They show the Cauchy distribution is relate to a certian Levy process: $$ |\mathbb{C}_1| \stackrel{\text{law}}{=} e^{\frac{\pi}{2} \hat{C}_1} \text{ with }\mathbb{E}\left[ e^{i\lambda \hat{C}_1} \right] = \frac{1}{\cosh \lambda} $$ If $X_t$ and $Y_t$ be $\mathbb{R}$-valued Brownian motions, then we have a $\mathbb{C}$-valued Brownian motion. $$ Z_t = X_t + i Y_t \text{ with }Z_0 = 1+0i$$ If we switch to polar coordinates, there is a "time-change" such that it is still Brownian motion, e.g. by conformal invariance of Brownian motion, as discussed in these early chapters in SLE.

- $\log R_t = \beta_{H_t} $ ($\log R_t$ is
**not**a Brownian motion) - $\theta_t = \gamma_{H_t} $

Then thy considered the hitting time for the Y-axis: $T = \text{inf} \{ t: X_t = 0 \}$. In the time-change parameterization: $$ H_T := T^{\gamma, *}_{\pi/2} $$ The $Y$-corrdinate at the hitting time $T$ (or $H_T$) is distributed as the Cauchy distribution.

\begin{eqnarray} \log | \mathbb{C}_1 | &\stackrel{\text{law}}{=}& \beta_{T^{\gamma, *}_{\pi/2}} \\ \frac{2}{\pi} \, \log | \mathbb{C}_1 | & \stackrel{\text{law}}{=} & \beta_{T^{\gamma, *}_{1}} \end{eqnarray}

The log-radius and angle $\beta$ and $\gamma$ are independent Brownian motions, over time. I was not able to varify this change of equations:

$$ \mathbb{E} \left[ e^{i\lambda \frac{2}{\pi} \log | \mathbb{C}_1|} \right] \stackrel{1}{=} \mathbb{E} \left[ e^{i\lambda \beta_{T^{\gamma,*}_{\pi/2}} } \right] \stackrel{2}{=} \mathbb{E} \left[ e^{ - \frac{\lambda^2}{2} T_1^{\gamma, *} } \right] \stackrel{3}{=} \frac{1}{\cosh \lambda} $$

I wish the authors would come up with different variable names for the hitting times.

**Why 2 and 3 are correct?** And can anyone explain the bigger picture of why these authors feel special values of zeta might be connected to these random processes?

Reference:

- Euler's formulae for $\zeta(2n)$ and products of Cauchy variables Paul Bourgade, Takahiko Fujita, Marc Yor