All Questions
10 questions
3
votes
1
answer
177
views
Mellin transform at $0$
Let $f$ be a smooth complex valued function with support on $[0,2]$ and with Mellin transform $\tilde{f}$. Recall that the Mellin transform is defined as $$\tilde{f}(w)=\int_0^{\infty}f(x)x^{w-1}dx.$$ ...
user543269
1
vote
0
answers
113
views
Are there any known statistics on the sign of the Stieltjes Constants?
The Stieltjes Constants $\gamma_n$ arise from considering the laurent series of the Riemann Zeta function at $s=1$
$$ \zeta(s) = \frac{1}{s-1} + \sum_{n=0}^{\infty} (-1)^n \frac{\gamma_n}{n!} (s-1)^n $...
- 2,689
5
votes
1
answer
427
views
Lindelöf hypotheses for derivatives of zeta
The Lindelöf hypothesis says that if we have:
$$\zeta(\sigma+iT)=\mathcal O(T^a)$$
Then if one considers $\sigma=1/2$ then $\inf a=0$. Further, from convexity and the functional equation this implies ...
- 83
3
votes
0
answers
105
views
Error function of the second moment of the divisor function
It is easy to show that the second moment of the divisor function has asymptotics:
$$\sum_{n\leq x} d_0(n)^2 = xP(\log(x))+E_2(x)$$
Where $P$ is some polynomial and that:
$$E_2 = o(x)$$
Previously, ...
- 83
7
votes
2
answers
720
views
On the asymptotic behaviour of the series $\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$ near $s=0$
I am interested in determining the behaviour of the the series/function
$$f(s)=\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$$
near $s=0$. It is clear that $f(0)$ is undefined....
- 405
0
votes
1
answer
164
views
How to check if you have the asymptotic solution of some equation? [closed]
Suppose I have an analytic function $f : \mathbb{R} \to \mathbb{R}$ and I have the asymptotic expansion of some $x_0$ up to a few terms in terms of $\epsilon$ for some $\epsilon \to 0$ which I believe ...
- 3
5
votes
1
answer
290
views
Overconcentration of Poles on the Circle of Convergence of a Power Series with Bounded Coefficients
Let $V$ be an arbitrary set of infinitely many positive integers, and let: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$$ Let $T_{V}$ denote the set of all $t\in\left[0,1\...
- 1,284
0
votes
2
answers
339
views
Error term in França-LeClair approximation of zeta zeros
The imaginary part of the $n$th critical zero of the Riemann zeta function with positive imaginary part (in increasing order) is asymptotically
$$
t_n \sim 2\pi\frac{n}{\log n}
$$
and has been ...
- 9,114
6
votes
1
answer
4k
views
About the logarithmic derivative of the Riemann zeta function
Let $\rho=\beta+i\gamma$ a non-trivial zeros of the Riemann zeta function and $s=\sigma+it$ a complex number. It is possible to prove that $$\frac{\zeta'}{\zeta}\left(s\right)=\sum_{\left|t-\gamma\...
- 219
1
vote
0
answers
78
views
Related to derivative of Modified Bessel I function wrt the order
I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$.
Using maple, it seems that $Re(\dfrac{I'...
