12 votes
Accepted

Lindelöf hypotheses for derivatives of zeta

The Lindelöf hypothesis yields the same bound for each derivative of $\zeta(s)$ via Cauchy's formula. Indeed, let $n\in\mathbb{N}$, $\sigma\in\mathbb{R}$, $T\in(1,\infty)$, $\varepsilon\in(0,1/2)$. ...
GH from MO's user avatar
  • 97.7k
10 votes
Accepted

Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$

As Henri Cohen remarked, the identity to prove is equivalent to $$\sum_{n=1}^\infty \frac{H_n^{(2)}}{(n+1)(2n+1)\binom{2n}{n}}=\frac{\pi^4}{972}.\tag{1}$$ In turn, this follows readily from the OP's ...
GH from MO's user avatar
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10 votes
Accepted

A real-valued analogue of the Weierstrass $\wp$ Function

Sums of this kind are called Epstein Zeta function. More generally, they are studied under the name of Lattice Sums. There is the book Lattice sums then and now devoted to this subject. There one can ...
Cave Johnson's user avatar
  • 5,397
8 votes
Accepted

Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$

There are counterexamples. Indeed, assume that the claimed bound holds for $x=1$. Then the Taylor series of $h(z)$ around $z=1$ converges in the disk $D=\{z:|z-1|<1\}$, hence $h(z)$ is analytic in $...
GH from MO's user avatar
  • 97.7k
6 votes

Nontrivial invariant transformations for heat equations

Yes, how about the Appell transform, see here. $$v(t,x)\mapsto\Gamma(t,x)v(-\frac1t,\frac xt)$$ where $\Gamma$ is the heat kernel. Truly nontrivial if you ask me. Of course, this inverts time and not ...
Funktorality's user avatar
5 votes

Zeroes of entire function on $\mathbb C^n$

This follows immediatelly from the following paper: The Zero Set of a Real Analytic Function
Nick S's user avatar
  • 1,990
4 votes

Bounds for analytic circles

The functions $\xi(s)$ and $$f(s):=\xi\left(\frac{s-i}{2}\right) + \xi\left(\frac{i+s}{2}\right)$$ grow faster than exponentially on the positive axis, hence they do not satisfy the first bound. This ...
GH from MO's user avatar
  • 97.7k
4 votes

Mellin-Barnes integral representation of the exponential function with a non-real argument

Might be helpful to accentuate the poles by expressing the M-B contour integral in more standard notation as $$ e^{-p \cdot x} = \frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} \frac{\pi}...
Tom Copeland's user avatar
  • 9,897
3 votes
Accepted

Bound for the $n$-th derivative of a proper rational function with no poles on the right half-plane

I will address the second, more general question (the answer implies the answer to the first one). Let us write the rational function as $p/q$, and assume for simplicity that zeros of $q$ are simple, ...
Alexandre Eremenko's user avatar
3 votes
Accepted

Nontrivial invariant transformations for heat equations

At least when $n=1$, there are no nontrivial transformations of this kind. Indeed, suppose that $v(t,x)=u(\tau(t,x),\xi(t,x))$, where $u_t=u_{xx}$. Then $u_{tx}=u_{xxx}$ and $u_{tt}=u_{xxxx}$, so that ...
Iosif Pinelis's user avatar
1 vote
Accepted

Bounds for analytic circles

It looks like @boJonson made a typo, mixing up $\xi(s)$ with $\Xi(t)$, defined, for $s=1/2+it$, as: $$\Xi(t) = \frac{1}{2} s(s-1) \pi^{-\frac{s}{2}} \Gamma\left(\frac{s}{2}\right) \zeta(s)$$ All the ...
Felixson's user avatar
  • 152
1 vote

Problem in understanding maximum principle for subharmonic functions

While this particular case is easier (as shown by Arnab), it also follows from the very useful, and not all that well known Katětov–Tong insertion theorem stating that: If $X$ is a normal topological ...
mrf's user avatar
  • 151
1 vote

Problem in understanding maximum principle for subharmonic functions

Proposition $:$ If $f$ is a u.s.c. function on $\Omega$ and bounded above, then there is a sequence $f_1 \geq f_2 \geq \cdots$ of continuous functions on $\Omega$ that are bounded above and that ...
Arnab Chattopadhyay.'s user avatar

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