All Questions
Tagged with arithmetic-functions asymptotics
7 questions
9
votes
2
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740
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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
2
votes
1
answer
281
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sum over primes involving divisor function (variation of the Titchmarsh divisor problem)
This question was also asked on MSE.
Does there exist an asymptotic estimate for the following sum over primes
$$
\sum_{p\leq x} \frac{\tau(p-1)}{p}\;,
$$
where $\tau(n)=\sum_{d|n}1$ is the divisor ...
- 215
2
votes
0
answers
110
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On variations of a claim due to Kaneko in terms of Lehmer means
This post is cross posted from Mathematics Stack Exchange, due that there was a mistake from my part (see the excellent partial answer and my thread of edits of my question on MSE) this post on ...
1
vote
1
answer
96
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On the behaviour for the quotient involving Fermat numbers of $\frac{\psi(F_m)}{F_m}$ where $\psi(x)$ denotes the Dedekind psi function
In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
0
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0
answers
68
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Around similar inequalities than an inequality due to Nicolas, that involve products of consecutive Ramanujan primes
This is cross-posted (and this post is a version to ask just around the veracity of Conjecture 1) as the post with identifier 3594907 and same title), that I've edited on Mathematics Stack Exchange ...
0
votes
0
answers
89
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A similar inequality for the Dedekind psi function, than an inequality stated by Schinzel
I would like to ask about the next question that seems to me interesting. I know an article that was written by Andrzej Schinzel in which he stated Lemma 2. In this post we denote the Dedekind psi ...
0
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0
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On the equation that involves the Dedekind psi function $\psi(x)=n$ with unique solution $x$, for a fixed integer $n\geq 1$
The Dedekind psi function is defined for a positive integer $m>1$ as
$$\psi(m)=m\prod_{\substack{p\mid m\\p\text{ prime}}}\left(1+\frac{1}{p}\right)\tag{1}$$
with the definition $\psi(1)=1$. See ...