# Questions tagged [arithmetic-functions]

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94
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### Strange lacunary Lambert series related to the Liouville function

Although I have my own interest for the Liouville function, I will suppress it here as the question seems to be interesting in its own right.
It occurred to me when I saw an answer by GH from MO to ...

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110 views

### Polynomials for the indicator function

The (one-variable) indicator function (or characteristic function) is defined as
$f_{t^*}:\mathbb{Z}_q\to \mathbb{Z}_q$ satisfying that
$f_{t^*}(t)=1$ if $t=t^*$ and $f_{t^*}(t)=0$, otherwise. (Here $...

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386 views

### Prove that two functions are equal only when $s \equiv \pm r^{\pm 1} \pmod{q}$

Let us fix a positive integer $q$, and let us define a functions $P: \mathbb{Z}\times \mathbb{N} \to \mathbb{Z}$ as follows:
$$ P(s,t) := \sum_{j=1}^t \left\lfloor \frac{j (s-1) + t}{q} \right\rfloor$$...

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260 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 $i=p^{m}$ ...

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101 views

### What is known about absolute convergence of Dirichlet inverses?

Given an arithmetic function $f$ such that the partial sums $\sum_{n \leq x} |f(n)|$ converge as $x$ approaches $\infty$, are there any results concerning the convergence properties of the series of ...

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47 views

### The number of elements with order less than $k$ in a larger cyclic group

I am working on a problem where it has become important to count (or at least bound from above and below) the number of elements of ${\bf Z}/n{\bf Z}$ that have order less than a given $k$, where $2\...

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188 views

### Are the numbers $\varphi(n^2)\sigma(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct?

Let $\varphi$ be Euler's totient function, and let $\sigma(n)=\sum_{d\mid n}d$ for $n=1,2,3,\ldots$. Both $\varphi$ and $\sigma$ are multiplicative functions. It is easy to see that the numbers
$$\...

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153 views

### Sign changes of a sequence

Let $f$ be an arithmetical function. Suppose that $f(n)>0$ if $n$ is in an integer set $A$ and that $f(n)<0$ for another integer set $B.$ Is there a result from number theory or an elementary ...

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284 views

### Where can I find the problem by Lagarias?

Jeffrey Lagarias proved, unconditionally, that:
$$
\sigma(n)<H_n+2\exp(H_n)\log(H_n)\qquad n>1
$$
This was posed as a problem in:
J. C. Lagarias, Problem 10949: A generous bound for divisor ...

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541 views

### $\frac{\sigma(n)}{n} < e \ln \ln (n)$ is true?

In Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187–213 (pdf)
we find the following result:
If the Riemann hypothesis is true ...

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127 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 ...

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191 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 $...

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352 views

### Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?

This question was posted in MSE in early August 2020. It did garner several upvotes, but did not receive any responses. I have therefore cross-posted it here, hoping that it gets answered.
Let $\...

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144 views

### Is $T(n)=\sum_{k=1}^{n}\frac{\lambda(k)\Lambda(k)}{k} \geq 0$ and what is the upper bound of $T(x)=\sum_{n\leq x} \lambda(n)\Lambda(n) $?

Let $\Lambda(n)$ denote the von Mangoldt function: $\Lambda(n)=\log p$ when $n=p^e$ is a prime power ($e\ge 1$) and $\Lambda(n)=0$ otherwise.
and $\lambda(n)$ be Liouville Function, , I'm interested ...

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84 views

### Arithmetic expansion of harmonic sum

Note: I have modified the initial question as follows:
Let $w_1, w_2, \ldots, w_d$ be positive weights, and $x_1, x_2, \ldots, x_d$ be positive variables. Now, let us consider the following harmonic ...

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569 views

### Is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound?

Pillai showed in 1929 that the function $A(n)$ giving the number solutions of the equation $\phi(x)=n$ is unbounded in (S. Pillai, On some functions connected with $\varphi(n)$, Bull. Amer. Math. Soc. ...

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169 views

### On the density map of the abundancy index

Let $σ$ be the sum-of-divisors function. Let $σ(n)/n$ be the abundancy index of $n$. Consider the density map $$f(x) = \lim_{N \to \infty} f_N(x) \ \ \text{ with } \ \ f_N(x) = \frac{1}{N} \#\{ 1 \...

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90 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 ...

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65 views

### Write large $n$ as $n_1+\ldots+n_k\ (n_1<\ldots<n_k)$ with $\varphi(n_1),\ldots,\varphi(n_k)\in\{x^k:\ x\in\mathbb Z\}$

Let $\varphi$ denote Euler's totient function.
QUESTION. Is it true that for each positive integer $k$ large integers $n$ can be written as $n_1+\ldots+n_k$ with $n_1,\ldots,n_k$ distinct positive ...

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85 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 ...

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142 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=...

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165 views

### When is $\phi(a^n+b^n+c^n)=0\mod n$?

A corollary Zsigmondy's Theorem leads to the following congruence (one can look to $(24)$),$\phi(a^n+b^n)=0\mod n$ whenever $a, b$ are coprime and $n \neq 2$ and $(a,b)\neq(1,1)$. (Here $\phi$ is the ...

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134 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 ...

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308 views

### Is there a smallest $r$ such that $n+\varphi(n)=\displaystyle \prod_{i=1}^r q_i$ always has solutions for mutually different odd primes $q_i $?

While discussing with Peter in one of the chatrooms on MSE I proposed an idea to try to find smallest natural number $r$ such that $n+\varphi(n)=\displaystyle \prod_{i=1}^r q_i$ has solutions for ...

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75 views

### Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^{\frac{{p_1}-3}{p_2}}-1)\equiv 0 \bmod p_1$ with $p_1,p_2\equiv 3\bmod 4$?

let $p_1$ and $p_2$ be positive primes such that $p_1,p_2 \equiv 3\bmod 4$
and $\phi$ is the Euler totiont function , I want to find the Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^...

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222 views

### If some numbers satisfy this divisibility condition with $\sigma$ and $\varphi$, are they necessarily multiples of $6$?

After doing some computations of the divisibility of $\sigma(n)$ by $n+ \varphi(n)$, mostly with Peter´s help, we found these solutions:
$n=2, 456, 828, 7584 ,33462 , 1357440, 1596048 ,1964544 ,...

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390 views

### Is this arithmetic function strictly positive and unbounded?

As requested by Mathphile, since there have been efforts but no complete solutions to some questions raised when this question was asked on MSE, and since we think that here the question is more ...

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335 views

### How to explain this property of totient?

I am running a program to search for solutions of $$\varphi(pm+1)=\varphi(pm+p+1).$$
So far, for $m=1,\ldots,327$ solutions have been found (some relatively large).
(in the body of the question, $p$ ...

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133 views

### Given $\,m=\prod_k {p_k}^{\alpha_k}\,$ and the function $\,g(m)=\sum_k \alpha_k(p_k-1)^2$, find all solutions of the equation $\,g(2n)=n$

Let's consider the unique decomposition of a natural number $\,m\,$ into its prime factors:
$$\prod_k {p_k}^{\alpha_k}$$
Then, let's define the following arithmetic function (completely additive) $\,g:...

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182 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 ...

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50 views

### On a type of equations that involve certain multiplicative functions and polynomials, in relation to their number of solutions

Past weekend I was interested in the sequence A058891 from the On-Line Encyclopedia of Integer Sequences, from this, inspired by the equation due to Benoit Cloitre (2002) that shows the comments, I ...

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196 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]) ...

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88 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 ...

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551 views

### Generalization of a problem, involving radicals and the floor function, proposed by Ramanujan to the Journal of the Indian Mathematical Society

The section Solved problems from the Wikipedia Floor and ceiling functions shows several problems proposed by Ramanujan ([1]). The purpose of this post, if possible, is try to get the generalization ...

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160 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(...

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419 views

### A curious conjecture: $\{\varphi(m^2)/\varphi(n^2):\ m,n=1,2,3,\ldots\}=\{r>0:\ r\in\mathbb Q\}$

Let $\varphi$ denote Euler's totient function. It is easy to see that all those numbers
$$\varphi(n^2)=n\varphi(n)\ \ (n=1,2,3,\ldots)$$
are pairwise distinct.
I have the following surprising ...

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84 views

### $\varphi(m+n)\mid n$ for some positive integer $n$

Let $\varphi$ be Euler's totient function.
If $p$ is a prime, then $\varphi(1+n)=n$ for $n=p-1$.
Question. Is it true that for each integer $m>1$ there is a positive integer $n\le m^2-m$ such that ...

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218 views

### Can we write each positive integer as $x^2+y^2+\varphi(z^2)$?

As odd squares are congruent to $1$ modulo $8$, any integer of the form $4^k(8m+7)$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$ cannot be written as the sum of three squares.
To avoid such congruence ...

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760 views

### Is the sum $\sum_{d\mid n}\frac1{d+1}$ never integral?

Recall that a positive integer $n$ is a perfect number if and only if
$$\frac{\sigma(n)}n=\sum_{d\mid n}\frac1d=2.$$
QUESTION: Is my following conjecture true?
Conjecture. (i) We have $\sum_{d\mid ...

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77 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 ...

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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. ...

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995 views

### Euler's totient function and Riemann hypothesis

I am looking for an upper-bound of the Euler's totient function $\varphi$ which would be equivalent to the Riemann hypothesis (RH). There is the following Nicolas' criterion about primorial numbers $...

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123 views

### Is there a way to reduce this problem to two variables through functions coming from arithmetic?

Consider following diophantine equation in $\mathbb Z[x,y,z]$ in three integer variables $x,y,z$
$$x^2+L(y,z)x+L_1(y)L_2(z)=0$$ where $L(y,z)$ is a non-homogeneous linear polynomial in $y,z$ and $L_1(...

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428 views

### A conjecture regarding odd perfect numbers

(Note: I asked this question in MSE this June 2018 but did not receive any responses there. I have therefore cross-posted it here, hoping that it gets answered.)
Let $\sigma(z)$ denote the sum of ...

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**1**answer

371 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.

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153 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 ...

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479 views

### Reference for inequality for $\sum\limits_{d \mid n}\frac{\log d}{d}.$

Let $f(n)=\sum\limits_{d \mid n}\frac{\log d}{d}.$
It is not hard to see that $f(n)\ll(\log\log n)^2$. Is there any reference for this inequality?
EDT 1: A possible answer is Analysis of the ...

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161 views

### Primes approximated by eigenvalues?

Let the matrix $T$ be defined by:
$$\displaystyle T(n,k) = -\varphi^{-1}(\operatorname{GCD}(n,k))$$
where $\varphi^{-1}$ is the Dirichlet inverse of the Euler totient function.
$$\varphi^{-1}(n) = \...

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365 views

### On $\varphi(m)\varphi(n)\equiv0\pmod{m+n}$

Euler's totient function $\varphi$ is multiplicative, and it plays important roles in number theory.
QUESTION: Is it true that for each integer $m>6$ we have $\varphi(m)\varphi(n)\equiv0\pmod{m+n}$...

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89 views

### Existence of equation about the product of the divisor sum function

Let $\sigma_k(n)$ be the sum of the $k$-th powers of the positive divisors of $n$ and $\mu(n)$ be the Möbius function.
As Arithmetic function - Wikipedia mentioned, there is an equation that $$\...