All Questions
Tagged with dirichlet-series asymptotics
15
questions
3
votes
1
answer
128
views
Power series for $\sum_{n=0}^\infty(-1)^n/n!^s$ (around $s=0$)
I'm looking for ways to compute the coefficients of the power series
$$
\sum_{n=0}^\infty\frac{(-1)^n}{n!^s}=\sum_{k=0}^{\infty}c_k s^k
$$
(a prior version of the question asked whether such an ...
- 40.1k
0
votes
0
answers
15
views
How fast does the proportion guaranteed by dirichlet converge?
I'm working on a counting problem and I'm using Dirichlets theorem (weak form) at some point in the counting. The problem is I don't know how fast something converges and I'm not very knowledgeable in ...
- 899
2
votes
1
answer
83
views
How to compute constants in asymptotic density of numbers divisible by subset of primes
I'm interested in the asymptotic density of the set $S$ of natural numbers divisible only by primes $p \equiv 1 \bmod 4$ (and similar subsets of $\mathbb{N}$). I'm aware of results which show that the ...
- 499
1
vote
0
answers
51
views
Why are these numbers close to $-\log(2)+\text{integer}\,i\pi$?
The following function $f(n)$ has been derived from the Dirichlet eta function:
$$f(n)=\log \left(\sum _{k=1}^n (-1)^{k+1} x^{c \log (k)}\right)-c \log (n) \log (x) \tag{$\ast$}$$
Let: $$s=\rho _1$$ ...
- 7,448
0
votes
0
answers
70
views
Questions on summatory function related to non-integer-powers
Consider the summatory function
$$f(x)=\sum\limits_{n=1}^x 1_{n\ne k^m}\tag{1}$$
where $1_{n\ne k^m}$ is the non-integer-power indicator function which returns $1$ when $n$ is a non-integer-power and $...
- 7,631
3
votes
0
answers
194
views
The average order of the divisor functions ${\sigma _\alpha }(n)$, where $\alpha < 0$ (Apostol, Intro to Analytic Number Theory, p.61)
In Apostol’s book, Theorem 3.6 (p.61) states a result concerning the average order of ${\sigma _\alpha }(n)$, where $\alpha < 0$
I am including an outline of Apostol’s approach, I hope I have ...
- 147
4
votes
1
answer
220
views
Asymptotic growth of summation of $\sum_{j\leq t}' 2^{\omega(j)}$ (restricting sum to a certain subset of $j$)
The following question came up in a thesis discussion I had with a student (undergraduate). I am a number theory researcher, but not within analytic number theory. One challenge I have faced is not ...
- 43
5
votes
1
answer
134
views
Asymptotic Expression for $ f(z) = z+ z^\frac{1}{2}+ z^\frac{1}{3}+ z^\frac{1}{4} +\dots + z^\frac{1}{N}$ with complex $z$?
Question (corrected)
I managed to prove:
$$ f(z) \sim \left\{
\begin{array}{ll}
- \ln |z| \int_0^{\frac{-N}{\ln|z|}} e^{-\frac{1}{|y|}} dy & |z|<< 1 \\
? & |z| \approx 1 \\
...
- 1,177
12
votes
1
answer
231
views
An asymptotic series for $\small\prod_{k=n}^\infty\operatorname{sinc}\left({2^{-k}}\pi\right),\,n\to\infty$
Using empirical methods, I conjectured that$^{[1]}$$\!^{[2]}$
$$\small\prod_{k=n}^\infty\operatorname{sinc}\left({2^{-k}}\pi\right)=1-\frac{2\pi^2}9\,4^{-n}+\frac{38 \,\pi ^4}{2025}\,4^{-2n}-\frac{...
- 47.3k
2
votes
0
answers
79
views
What about the convergence of $\lim_{x\to\infty}\sum_{1\leq n\leq x}\frac{\lambda(n)}{\operatorname{rad}(n)}\log\left(\frac{x}{n}\right)$?
Let $\operatorname{rad}(n)$ the product of distinct primes dividing $n>1$, with the definition $\operatorname{rad}(1)=1$ (if you need it, see the definition of the radical of an integer from this ...
user243301
0
votes
0
answers
74
views
Asymptotic behaviour of a Dirichlet series as t goes to infinity Titchmarsh the theory of functions
I am stuck on a proof from a proposition on Titchmarsh's book "The Theory of Functions", at page 297 which says that for a Dirichlet series we have: $f(s)=O(|t|^{1-(\sigma-\sigma_0)+\epsilon})$ as $|t|...
- 697
3
votes
0
answers
116
views
A Wiener-Ikehara variant with higher order poles
The problem: I am concerned of getting a generalisation of the Wiener-Ikehara theorem for Dirichlet series which are analytic in the plane $\{s\in\mathbb{C}:\sigma>1\}$ and extend analytically over ...
- 1,742
6
votes
1
answer
283
views
Average Order of $\frac{1}{\mathrm{rad}(n)}$
Again a question about $\mathrm{rad}(n).$
Let $\mathrm{rad}(n)$ denote the radical of an integer $n$, which is the product of the distinct prime numbers dividing $n$. Or equivalently, $$\mathrm{rad}(...
user261687
8
votes
1
answer
715
views
Average order of $\mathrm{rad}(n)$
Let $\mathrm{rad}(n)$ denote the radical of an integer $n$, which is the product of the distinct prime numbers dividing n. Or equivalently, $$\mathrm{rad}(n)=\prod_{\scriptstyle p\mid n\atop p\text{ ...
user261687
10
votes
2
answers
927
views
Approximation of Products of Truncated Prime $\zeta$ Functions
The problem arose, while I was looking at products of power prime zeta functions
$$
P_x(ks)=\sum_{p\,\in\mathrm{\,primes}\leq x} p^{-ks},
$$
with $k\in \mathbb{N}$ and $s=it$ with real $t$.
By using (...
- 18.6k