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I am looking for reference for proof that $\Phi(t)$ is strictly decreasing for $t>0$ and the first derivative of $\Phi(t)$ is negative for $t>0$ (see Page 5 in Conrey's article below)

Conrey says $\Phi(t)$ is positive for positive $t$ and its first derivative $\Phi'$ is also positive. I think he means the first derivative $\frac{d}{dt} \Phi$ is negative for $t \in \left[0, \infty\right]$.

https://www.ams.org/notices/200303/fea-conrey-web.pdf#page=5

I am searching for reference for this proof or proof itself.

Any pointers much appreciated!

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    $\begingroup$ I don’t have time to check the details, but from memory, the argument for positivity of $\Phi$ goes something like this: if $t>0$ then the infinite series representation for $\Phi$ is easily seen to consist of nonnegative terms. And if $t<0$, use the fact that $\Phi$ is even to reduce to the previous case. (In the background, the even symmetry of $\Phi$ is a reformulation of the fact that $\Phi$ has a simple representation in terms of the Jacobi theta function $\\theta(x)$, which is a modular form of weight $1/2$.) $\endgroup$
    – Dan Romik
    Commented Apr 9, 2022 at 3:24
  • $\begingroup$ (The argument for $\Phi’$ should be essentially the same I think.) $\endgroup$
    – Dan Romik
    Commented Apr 9, 2022 at 3:24
  • $\begingroup$ For $t>T$ large enough the sign of $\phi'(t)$ is obvious, so it remains numerically what happens (for $\phi'(t)/t=\ldots$) on $[0,T]$. $\endgroup$
    – reuns
    Commented Apr 9, 2022 at 13:39

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