All Questions
Tagged with analytic-number-theory arithmetic-functions
16 questions with no upvoted or accepted answers
11
votes
0
answers
238
views
Strong uniqueness of Euler's totient function
Let $f:\mathbb N\to \mathbb C$ be some arithmetical function. Define $\varphi_f(n)$ by the following formula:
$$
\varphi_f(n)=\sum_{\substack{k\leq n \\ (k,n)=1}}f(k).
$$
In other words, $\varphi_f(...
- 7,193
4
votes
0
answers
151
views
Identities to go from $\sum_{n\leq x} \mu(n) \log \frac{x}{n}$ to $M(x) = \sum_{n\leq x} \mu(n)$?
Let $\mu$ be the Möbius function. Say we have a bound on $\check{M}(x) = \sum_{n\leq x} \mu(n) \log \frac{x}{n}$ of the form $|\check{M}(x)|\leq \epsilon x$ for all $x\geq x_0$.
It is then easy to ...
- 20.2k
4
votes
0
answers
86
views
On Carmichael function and aliquot parts of odd perfect numbers
I've asked nine months ago this question on Mathematics Stack Exchange with identifier 4430381 and same title. There is not answer for this question on Mathematics Stack Exchange, I wondered if this ...
4
votes
0
answers
81
views
Joint mean values of arithmetic functions in sequences and families of sequences
This is a bit of a follow up question to this question I asked a couple days ago. The main content of that post can be phrased as asking for a nontrivial lower bound on the sum
$$
\sum_{n\leq x} \...
- 1,990
4
votes
0
answers
413
views
Maximal order of Hooley's Delta function?
There is a large literature on Hooley's
$$
\Delta(n)=\max_u\sum_{d|n,\ e^u\le d< e^{u+1}}1
$$
giving its normal and average order. What is known of its maximal order?
Clearly $\Delta(n)\le d(n)$ ...
- 9,114
2
votes
0
answers
357
views
Mertens Bound and the Riemann Hypothesis
Let $M(x)$ denote the Mertens function ($M(x)=\sum_{i=1}^{x}\mu(i)$ where $\mu(i)$ is the Möbius function) and let $\Lambda(i)$ denote the Mangoldt function ($\Lambda(i)$ equals $\log(p)$ if $i=p^{m}$ ...
- 177
2
votes
0
answers
110
views
On variations of a claim due to Kaneko in terms of Lehmer means
This post is cross posted from Mathematics Stack Exchange, due that there was a mistake from my part (see the excellent partial answer and my thread of edits of my question on MSE) this post on ...
2
votes
0
answers
117
views
Alternating series of arithmetic function values
Some time ago I asked about alternating sums like
$\sum_{n\leq x}(-1)^n \phi(n)$ where $\phi(n)$ is Euler's totient
and the similar sum involving $\sigma(n)$ where $\sigma(n)=\sum_{d|n}d.$
Certain ...
- 112
1
vote
0
answers
165
views
Relationship between two types of partition functions
Referring to this unanswered question on MS, I'm posting the same question here:
For $s\in \mathbb{C},\Re(s)>1 $, consider:
$$\prod_{k=1}^{\infty}\prod_{n=2}^{\infty}\frac{1}{1-n^{-ks}}= \prod_{k=1}...
- 181
1
vote
0
answers
167
views
On two formulas involving the $k$-fold divisor function $d_k$ and the function $r_k$
I have a puzzle which needs some help form the experts here. Let $d(n)$ be the divisor function, and $d_k(n)$ the $k$-fold divisor function.
I) It is known that, for any positive integer $h$,
$$d(n+h)...
- 563
1
vote
0
answers
90
views
An arithmetic function involving arbitrary (but fixed) number of divisors
I need at least basic information about generating functions of the following class of arithmetic functions, grouped by levels $k$.
Fix some $k$ and some family $\varepsilon_*=(\varepsilon_\sigma)_{\...
- 18.4k
0
votes
0
answers
193
views
Possible implications of the bound $\sum_{n\leq x}\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)=O\left(x\right)$
Let $\lambda(n)$ be the Liouville function and consider the Goldbach-type problem $\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)$. Assume the Riemann hypothesis, the ...
- 1
0
votes
0
answers
68
views
Around similar inequalities than an inequality due to Nicolas, that involve products of consecutive Ramanujan primes
This is cross-posted (and this post is a version to ask just around the veracity of Conjecture 1) as the post with identifier 3594907 and same title), that I've edited on Mathematics Stack Exchange ...
0
votes
0
answers
89
views
A similar inequality for the Dedekind psi function, than an inequality stated by Schinzel
I would like to ask about the next question that seems to me interesting. I know an article that was written by Andrzej Schinzel in which he stated Lemma 2. In this post we denote the Dedekind psi ...
0
votes
0
answers
154
views
On the equation that involves the Dedekind psi function $\psi(x)=n$ with unique solution $x$, for a fixed integer $n\geq 1$
The Dedekind psi function is defined for a positive integer $m>1$ as
$$\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 ...
0
votes
0
answers
219
views
On an inequality involving the Lambert $W$ function and the sum of divisors function
Let $W(n)$ be the principal/main branch of the Lambert $W$ function (this is the Wikipedia related to this special function). I was inspired in Robin equivalence to the Riemann hypothesis (see [1]) ...