# Bounding the derivative of H t

We continue using the notation from Effective bounds on H_t - second approach. We also assume $T \geq 10$ (so $x \geq 20$).

Since

$H_t(z) = \frac{1}{8} \xi_t( s)$

with $s := \frac{1+iz}{2}$, we have

$\frac{d}{dz} H_t(z) = \frac{i}{16} \frac{d}{ds} \xi_t(s).$

Next, we have

$\xi_t(s) = \sum_{n=1}^N F_{t,n}(s) + F_{t,n}(1-s) + G_{t,N}(s) + G_{t,N}(1-s)$

(using the convention $F(\bar{s}) = \bar{F(s)}$ for $s$ in the lower half-plane). Thus (assuming that we are not at a discontinuity for $latex N$) we have

$\frac{d}{ds} \xi_t(s) = \sum_{n=1}^N F'_{t,n}(s) - F'_{t,n}(1-s) + G'_{t,N}(s) - G'_{t,N}(1-s).$

Now we have for any $\alpha_n$ that

$F_{t,n}(s) = \exp( - \frac{t}{4} \alpha_n^2 ) \int_{-\infty}^\infty \exp( - \sqrt{t} \alpha_n u) F_{0,n}( s + \sqrt{t} u + \frac{t}{2} \alpha_n ) \frac{1}{\sqrt{\pi}} e^{-u^2}\ du,$

hence in differentiation under the integral sign (justifiable for instance using the Cauchy integral formula and Fubini's theorem)

$F'_{t,n}(s) = \exp( - \frac{t}{4} \alpha_n^2 ) \int_{-\infty}^\infty \exp( - \sqrt{t} \alpha_n u) \frac{\partial}{\partial s} F_{0,n}( s + \sqrt{t} u + \frac{t}{2} \alpha_n ) \frac{1}{\sqrt{\pi}} e^{-u^2}\ du.$

This identity is true for any $\alpha_n$; we now set $\alpha_n = \alpha_n(s)$ as in the above wiki page. One can replace $\frac{\partial}{\partial s}$ on the RHS by $\frac{1}{\sqrt{t}} \frac{\partial}{\partial u}$ and integrate by parts to conclude that

$F'_{t,n}(s) = \exp( - \frac{t}{4} \alpha_n(s)^2 ) \int_{-\infty}^\infty \exp( - \sqrt{t} \alpha_n(s) u) F_{0,n}( s + \sqrt{t} u + \frac{t}{2} \alpha_n ) \frac{1}{\sqrt{\pi}} (\alpha_n(s) + \frac{2u}{\sqrt{t}}) e^{-u^2}\ du.$

We have

$F_{0,n}( s + \sqrt{t} u + \frac{t}{2} \alpha_n(s)) = H_{0,n}(s) \exp( (\sqrt{t} u + \frac{t}{2} \alpha_n(s)) \alpha_n(s) + O_{\leq}( \frac{1}{4 (T - 3.08)} ( |\sqrt{t} u + \frac{t}{2} \alpha_n(s)|^2 + \frac{2}{3} ) ) )$

and hence

$|F'_{t,n}(s)| \leq \exp( \frac{t}{4} \mathrm{Re}(\alpha_n(s)^2) ) |H_{0,n}(s)| \int_{-\infty}^\infty \exp( \frac{1}{4 (T - 3.08)} ( |\sqrt{t} u + \frac{t}{2} \alpha_n(s)|^2 + \frac{2}{3} ) ) ) \frac{1}{\sqrt{\pi}} (|\alpha_n(s)| + |\frac{2u}{\sqrt{t}}|) e^{-u^2}\ du.$

We can bound

$\frac{1}{4 (T - 3.08)} ( |\sqrt{t} u + \frac{t}{2} \alpha_n(s)|^2 + \frac{2}{3} ) \leq \frac{1}{2(T-3.08)} ( tu^2 + \frac{t^2}{4} |\alpha_n(s)|^2 + \frac{1}{3} )$

and also write

$\exp( \frac{t}{4} \mathrm{Re}(\alpha_n(s)^2) ) |H_{0,n}(s)| = \exp( \frac{t}{4} \mathrm{Re}(\alpha_1(s)^2) ) |H_{0,1}(s)| \frac{b_n}{n^{\mathrm{Re} s_A}}$
$= 8 |\lambda| |B^{eff}_0| \frac{b_n^2}{n^{\sigma + \frac{t}{2} \mathrm{Re} \alpha_1(s)}}$

to obtain

$\frac{1}{8} |F'_{t,n}(s)| \leq |\lambda| |B^{eff}_0| \frac{b_n}{n^{\mathrm{Re} s_A}} \frac{\exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} |\alpha_n(s)|^2 + \frac{1}{3}) ) }{\sqrt{\pi}} \int_{-\infty}^\infty (|\alpha_n(s)| + |\frac{2u}{\sqrt{t}}|) \exp( -(1-\frac{t}{2(T-3.08)}) u^2 )\ du.$

Noting that

$\int_{-\infty}^\infty e^{-a u^2}\ du = a^{-1/2} \sqrt{\pi}$

and

$\int_{-\infty}^\infty e^{-a u^2} |u|\ du = a^{-1}$

for any $a\gt0$, we thus have

$\frac{1}{8} |F'_{t,n}(s)| \leq |\lambda| |B^{eff}_0| \frac{b_n}{n^{\mathrm{Re} s_A}} \exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} |\alpha_n(s)|^2 + \frac{1}{3}) ) ( |\alpha_n(s)| (1-\frac{t}{2(T-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T-3.08)})^{-1}).$

We can bound $|\alpha_n(s)| \leq |\alpha_1(s)|$ and $T \geq T_N$ and conclude that

$\frac{1}{8} \sum_{n=1}^N |F'_{t,n}(s)| \leq |\lambda| |B^{eff}_0|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} |\alpha_1(s)|^2 + \frac{1}{3}) ) ( |\alpha_1(s)| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) S_{\mathrm{Re} s_A - \frac{t}{2} \log N, t}(N).$

where $S_{\sigma,t}(N)$ was defined in Estimating a sum. Similarly

$\frac{1}{8} \sum_{n=1}^N |F'_{t,n}(1-s)| \leq |B^{eff}_0|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} |\alpha_1(1-s)|^2 + \frac{1}{3}) ) ( |\alpha_1(1-s)| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) S_{\mathrm{Re} s_B - \frac{t}{2} \log N, t}(N).$