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 $n\in\mathbb{N}$, $\sigma\in\mathbb{R}$, $T\in(1,\infty)$, $\varepsilon\in(0,1/2)$. ...
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 ...
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 ...
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 $...
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 ...
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
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 ...
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
$$ e^{-p \cdot x} = \frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} \frac{\pi}...
3
votes
Accepted
Bound for the $n$-th derivative of a proper rational function with no poles on the right half-plane
I will address the second, more general question (the answer implies the answer to the first one). Let us write the rational function as $p/q$, and assume for simplicity that zeros of $q$ are simple, ...
3
votes
Accepted
Nontrivial invariant transformations for heat equations
At least when $n=1$, there are no nontrivial transformations of this kind.
Indeed, suppose that $v(t,x)=u(\tau(t,x),\xi(t,x))$, where $u_t=u_{xx}$. Then $u_{tx}=u_{xxx}$ and $u_{tt}=u_{xxxx}$, so that ...
1
vote
Accepted
Bounds for analytic circles
It looks like @boJonson made a typo, mixing up $\xi(s)$ with $\Xi(t)$, defined, for $s=1/2+it$, as:
$$\Xi(t) = \frac{1}{2} s(s-1) \pi^{-\frac{s}{2}} \Gamma\left(\frac{s}{2}\right) \zeta(s)$$
All the ...
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 ...
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 ...
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