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4 votes
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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 ...
H A Helfgott's user avatar
  • 20.2k
4 votes
1 answer
255 views

First occurrence of formula for $\sum_{n\leq x} \mu(n) \log n$ in terms of $\psi(y)-\lfloor y\rfloor$?

The identity contained in the last two displayed equations in the following passage (from page 110 in Ayoub's An Introduction to the Analytic Theory of Numbers, 1963) gives us right away a simple ...
H A Helfgott's user avatar
  • 20.2k
3 votes
1 answer
189 views

Brun-Titchmarsh for sum over square divisors

Let $f(n)$ be a nonnegative arithmetic function satisfying $f(p^l) \leq A_1^l$ for all primes $p$, integers $l\geq 1$, and some constant $A_1$; $f(n) \leq A_2 n^\varepsilon$ for all $\varepsilon > ...
Joshua Stucky's user avatar
4 votes
1 answer
175 views

Behavior of $m(x)\sqrt{x}$ where $m(x)=\sum_{n\leq x}\frac{\mu(n)}{n}$

Let $M(x) = \sum_{n\leq x} \mu(n)$ and $m(x) = \sum_{n\leq x} \frac{\mu(n)}{n}$, where $\mu(n)$ is the Möbius function. We know that (it is not the best known bounds): $$\limsup_{x \to \infty} M(x)x^{-...
 Babar's user avatar
  • 611
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 ...
Yep's user avatar
  • 1
11 votes
2 answers
1k views

A question on Euler's totient function

With reference to the Euler's totient function $\phi(\cdot)$, given any $n \in \mathbb{Z}^+$, it's quite straightforward to find $\phi(n)$. In contrast, given $n \in \mathbb{Z}^+$, even though there ...
Eureka's user avatar
  • 211
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}...
mohammad-83's user avatar
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)...
hofnumber's user avatar
  • 563
0 votes
1 answer
338 views

Estimating a sum involving the von Mangoldt function

I'd like to know the estimate of the following sum $$\sum_{n\leq x}\sum_{d|n}\Lambda(d)\frac{\phi(d)}{d} $$ where $\Lambda(d)$ is the von mangoldt function and $\phi(d)$ is the Euler totient function. ...
Beta's user avatar
  • 365
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 ...
user142929's user avatar
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 ...
user142929's user avatar
1 vote
1 answer
347 views

On equations with arithmetic functions [closed]

Is this good topic for research: equations with arithmetic functions, for example equations like $\varphi(n)=\sigma(n)$ or $\varphi(n)+\sigma(n)=d(n)$ ? If Anyone here have an advise please tell me ...
Omega's user avatar
  • 31
6 votes
1 answer
213 views

Upper bound on minimum number of prime factors in short intervals

Suppose that $H = H(X)$ is some quantity growing with $X$. Are there any bounds on $$F(X, H) = \min_{X < n\le X + H} \omega(n)?$$ It isn't hard to obtain a lower bound $\max_{x\sim X} F(X, H)\gg \...
Mayank Pandey's user avatar
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 ...
user142929's user avatar
1 vote
1 answer
96 views

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 ...
user142929's user avatar
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} \...
Joshua Stucky's user avatar
3 votes
1 answer
134 views

Is it possible to find an estimate of $\sum_{k=1}^n\frac1{\varphi(k\cdot p_k)}$?

Is it possible to find an estimate of the summation $$s(n)=\sum_{k=1}^n\frac1{\varphi(k\cdot p_k)}$$ where $\varphi(n)$ is the totient function and $p_k$ the k-th prime? The corresponding series seems ...
Augusto Santi's user avatar
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 $i=p^{m}$ ...
Sourangshu Ghosh's user avatar
3 votes
1 answer
154 views

Arithmetical function comparable to sine function [closed]

I was wondering if there exists or can we construct (using known arithmetic functions) an arithmetical function that has the same behaviour of the function sine or comparable to it (I mean that ...
Khadija Mbarki's user avatar
4 votes
1 answer
291 views

Generalization of the The Liouville Lambda function

Let $n=p^{\alpha_1}_1 \cdots p^{\alpha_m}_m,$ and define $$\lambda_k(n)= (-1)^{ [\frac{\Omega(n)}{k} ]},$$ where $\Omega(n)= \alpha_1 + \cdots + \alpha_k,$ and $[\cdot]$ is the floor function. For $...
Farzad Aryan's user avatar
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 ...
user142929's user avatar
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 ...
user142929's user avatar
0 votes
2 answers
390 views

What is the dirichlet series of $f(n)=\sum_{d | n}(\log d) / d$ function? [closed]

My opinion is ; We may use id(d)=d arithmetic function and log*id dirichlet convolution in the question. i thought that ; when we multiply and divide n with $(\log d) / d$ we obtain $F(S)=\sum_{n=...
user1062's user avatar
  • 105
2 votes
1 answer
198 views

Bounds for two arithmetic functions, when one assumes that $n$ are odd perfect numbers

For an integer $n>1$ in this post we denote the Dedekind psi function as $\psi(n)=n\prod_{\substack{p\mid n\\p\text{ prime}}}\left(1+\frac{1}{p}\right)$ and the product of distinct primes dividing ...
user142929's user avatar
7 votes
1 answer
231 views

The asymptotic of $|\{1\leq n\leq x|\gcd(n,S(n))=1\}|$, with $S(n)$ the sum of remainders, and get idea for other miscellany problem

Let $n\geq 1$ be an integer. In this post we denote the sum of remainders function as $$S(n)=\sum_{k=1}^n n \bmod k,$$ for example $S(1)=S(2)=0+0$ and $S(5)=0+1+2+1+0=4$. In the literature there are ...
user142929's user avatar
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]) ...
user142929's user avatar
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 ...
TPTW's user avatar
  • 112
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(...
Alexander Kalmynin's user avatar
5 votes
1 answer
180 views

Uniformity in Wirsing's Mean Value Theorems

In two important papers of Wirsing, namely "Das asymptotische Verhalten von Summen über multiplikative Funktionen" (1961) and its follow up (1967), several results on mean values of multiplicative ...
Ofir Gorodetsky's user avatar
6 votes
2 answers
2k views

Does this multiplicative function have a name? If so, what is known about it?

It is well-known that the Euler $\phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $\phi(mn) = \phi(m)\phi(n)$. Thus it is defined by its values on prime powers. ...
Stanley Yao Xiao's user avatar
8 votes
1 answer
426 views

Goldbach's conjecture for the Liouville function

Is it true that for every even integer $N > 2$, there exist positive integers $a,b$ such that $a + b = N$ and $\lambda(a) = \lambda(b) = -1$ ? Here $\lambda$ is the Liouville function.
Pablo's user avatar
  • 11.3k
4 votes
1 answer
170 views

The number of solutions of the equation $ax_1x_2+by_1y_2=n$

The equation $x_1x_2+y_1y_2=n$ is well-studied (Ingham, Heath-Brown, Deshouillers & Iwaniec, Ismoilov) because it arises in an additive divisor problem. The number of solutions in positive ...
Alexey Ustinov's user avatar
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)_{\...
მამუკა ჯიბლაძე's user avatar
5 votes
2 answers
314 views

Congruences for the non-divisors of Euler's $\phi(n)$

If $n$ is composite, then $\phi(n) < n-1$: hence, there is at least one number $d$ which does not divide $\phi(n)$ but divides$(n-1)$. We shall call $d$ the totient divisor of $n$. The purist will ...
Nilotpal Kanti Sinha's user avatar
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: ...
Vaclav Kotesovec's user avatar
0 votes
1 answer
475 views

Sum of digits of a power [closed]

Are there any explicit formula for a sum of digits for a power in the given base? A problem to be specific: find a sum of digits for a number $2^{100}$ in the system with a base 5. In the system with ...
Sergei's user avatar
  • 1,550
6 votes
1 answer
331 views

A question about $(0,1]$-valued multiplicative functions

Suppose $f:\mathbb{N}\to [0,1]$ is a multiplicative function (i.e. $f(nm)=f(n)f(m)$ whenever $m$ and $n$ are coprime). Suppose $f$ has non-zero mean, which means $$ \lim_{N\to\infty}\frac{1}{N} \sum_{...
Karl's user avatar
  • 63
1 vote
1 answer
163 views

estimate an sum

I need estimate the following sum: $\sum_{d=1}^{n}\frac{\mu(d)}{d}\sum_{k=1}^{\lfloor n/d\rfloor}\frac{1}{k}\frac{q^k}{1-q^{-kd}}$, where $q>1$ and $\mu$ is the Möbius function. To obtain the ...
Dianbin Bao's user avatar
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 $\Delta(n)\le d(n)$ ...
Charles's user avatar
  • 9,114
15 votes
2 answers
1k views

Sum of $\sum_{k=1}^nd(k^2)$

There is a literature dealing with $$ \sum_{k\le x}d(f(k)) $$ where $f$ is an irreducible polynomial and $d(n)$ is the number of divisors of $n$. Erdos 1952 shows that the sum $\asymp x\log x,$ which ...
Charles's user avatar
  • 9,114