All Questions
Tagged with analytic-number-theory asymptotics
117 questions
9
votes
2
answers
740
views
Asymptotics of product of Euler's totient function (A001088)?
Conjecture:
\begin{align}
\lim_{n\to \infty } \, \frac{\left(\prod _{k=1}^n \phi (k)\right){}^{1/n}}{n}\sim 0.2059\text{...}
\end{align}
The numerical result from 100000 terms is:
My questions are:
...
- 690
0
votes
1
answer
404
views
Asymptotics of "ugly" function elucidate Goldbach's conjecture?
Question
We now define the following "ugly" function:
$$ A_c(s,r,n,m) =
\begin{cases}
1 & \text{ if only $sr+nm=2c$ } \\ 0 & \text{otherwise}
\end{cases}
$$
How does the "ugly"...
- 89
2
votes
0
answers
110
views
Bounds on the number of zeros of a quadratic form
Let $Q(x_1, \dots, x_n)$ be a non-degenerate indefinite quadratic form with integer coefficients. Let $N(Q,T)$ be the set of vectors $x=(x_1, \dots, x_n) \in {\mathbb Z}^n$ such that $|x|<T$ and $Q(...
- 6,224
3
votes
0
answers
320
views
On sets of coprime numbers
We know that from prime number theorem that the number of primes below $n$ and above $\frac n2$ (denoted by $\pi_{n,\frac n2}$ is approximately $$\pi_{n,\frac n2}\approx\frac{n}{2\ln n}.$$
Denote by $...
user76479
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'...
11
votes
2
answers
771
views
Tauberian theorem with better error term
This is a fairly vague question.
Suppose we have a sequence of positive numbers $(c_n)_n$ and we want to find an asymptotic formula for $S(x) = \sum_{n \leq X} c_n$. In favorable circumstances, ...
- 1,362
2
votes
1
answer
338
views
Bombieri-Vinogradov in short intervals
In 1985 Perelli, Pintz & Salerno proved a short-interval form of the Bombieri-Vinogradov theorem with $\theta \in (7/12, 1]$. Have there been any improvements on this, in particular with the ...
- 338
4
votes
0
answers
306
views
Effective version of the Bombieri-Vinogradov theorem
Is there an effective version of the Bombieri-Vinogradov Theorem, in that have bounds on the implied constant been found?
- 1,890
25
votes
1
answer
2k
views
Wrong asymptotics of OEIS A000607 (number of partitions of an integer in prime parts)?
Sequence A000607 in the Online Encyclopedia of Integer Sequences is the number of partitions of $n$ into prime parts. For example, there are $5$ partitions of $10$ into prime parts: $10 = 2 + 2 + 2 + ...
- 690
2
votes
2
answers
590
views
Asymptotic formula for the number of ways to write a number as the sum of $k$ triangular numbers
How would one derive an asymptotic formula for the number of representations of a number $n$ as the sum of $k$ numbers of the form $\frac{m(m + 1)}{2}$
I think that one could use the circle method, ...
- 1,890
6
votes
1
answer
809
views
Probability that a positive integer is in the range of the Euler phi function
Define $f(n) = |\{m : m\le n, \exists k \text{ s.t. }\phi(k) = m\}|$.
Clearly, $f(n)\le \left\lfloor \frac{n}{2}\right\rfloor + 1$ since $\phi(n)$ is even for all $n > 2$.
Is $\limsup_{n\...
- 1,890
21
votes
1
answer
771
views
Covering a set with geometric progressions
Consider the set $S_n=\{1,2,\cdots ,n\}$. What is the minimum number of distinct geometric progressions that cover $S_n$? Let us call this number $a_n$. I was wondering about this number after doing a ...
- 1,090
20
votes
1
answer
1k
views
Quantitative lower bounds related to Zhang's theorem on bounded gaps
Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ n+h_{...
- 11.4k
4
votes
2
answers
517
views
Average involving the Euler phi function
Does
$$\frac{1}{N^2}\sum _{d=1}^N \log d \sum _{n=1}^{N/d} \frac{\phi(n)}{\log (dn)},$$
converges or not when $N$ goes to infinity?
- 43
6
votes
1
answer
548
views
Equidistribution on the unit circle of particular sequences of finite subsets
Given a strictly convex function $g : [0, 1] \to \mathbb{R}$, I'm curious about the asymptotic distribution of the points $\exp{(2 \pi i N g(n / N))}$ for $n = 1, 2, \dots, N$, counted with ...
3
votes
1
answer
516
views
About the asymptotics of LCM
Let $g(x,c)$ be a uniformly random integer in the range $(x,x+c)$ and $LCM[x_1,x_2...x_i]$ the lowest common multiple of the integers $x_i$.
A) Does the limit of (the asymptotics of $LCM[g(3^1,c),g(...
- 61
2
votes
3
answers
343
views
complete estimates of the error for a well-known asymptotic expression of partition p(n,m)
Let $p(n,m)$ be the number of partitions of an integer $n$
into integers $\le m$, we have a well-known asymptotic expression:
For a fixed $m$ and $n\to\infty$,
$$p(n,m)=\frac{n^{m-1}}{m!(m-1)!} (1+...
- 199