Questions tagged [arithmetic-functions]

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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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66 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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110 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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160 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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1answer
121 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 ...
2
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1answer
219 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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71 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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219 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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380 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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1answer
332 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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127 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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148 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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41 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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183 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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67 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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60 views

Prove that the Mertens function is invariant under matrix inversion for $n>3$

Consider the lower triangular matrix $T$ with the definition: $n \leq3 :$ $$T(n, 1)=1$$ $n>3:$ $$T(n,1)=x_n$$ $n \geq k:$ $n \leq 3:$ $$T(n,k)=1$$ $n \geq k:$ $n>3:$ $k=2 \text{ or } k=3:$ ...
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155 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(...
4
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1answer
378 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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74 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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214 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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1answer
608 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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61 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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933 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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120 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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1answer
380 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 ...
8
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1answer
358 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.
4
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1answer
151 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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1answer
435 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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145 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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1answer
352 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 $$\...
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1answer
250 views

The Euler's totient function and the product of distinct primes dividing $n$ versus the Heronian means

For integers $n\geq 1$ with $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p$$ we denote the squarefree kernel or radical of an integer $n$ (see if you want this Wikipedia). And $\...
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140 views

Is there a closed form for the discrete convolution of $\sigma_1$ and $\sigma_2$?

I am trying to find a closed form for the following sum: $$\sum_{k = 1}^{n-1}\sigma_1(k) \sigma_2(n-k)$$ where $\sigma_i$ is the sum of divisors and $\sigma_2$ is the sum of squares of divisors. ...
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3answers
423 views

Evaluating the sum of $kl^2$ over $p,q,k,l$ such that $pk +ql = n$

I have come across the following sum: $$\sum_{\substack{p, q, k, l \in \mathbb{N} \\ k > l \\ pk + ql = n}}kl^2$$ and I am trying to simplify it, hoping to get a nice formula in terms of $n$ and ...
4
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1answer
303 views

A special kind of multiplicative function $f: \mathbb N \to \mathbb N$ such that $f(p)=p+k$ for all odd prime $p$, where $k>1$ is a fixed odd integer

For which odd positive integer $k$, can we find a multiplicative function $f: \mathbb N \to \mathbb N$ satisfying the following conditions : $f(p)=p+k$ for all large enough odd prime $p$ and the set $...
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84 views

Name and theory of multi-valued functions $F:\mathbb{N}^k \rightarrow \mathbb{N}^l$

In computability theory there are considered mostly single-valued functions $f:\mathbb{N}^k \rightarrow \mathbb{N}$. (Let $\mathbb{N}$ be a placeholder for $\mathbb{N}$ or any initial segment $[0,n]$ ...
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1answer
136 views

Tighter upper bound for $\sum_{i=1}^kA_i\log(\frac{A_i}{e})$

What is the tightest upper bound one can obtain for the following expression $$\sum_{i=1}^kA_i\log(\frac{A_i}{e})$$ subject to $\sum_{i = 1}^k A_i = C$ in terms of $C$ and $k$? A very loose upper ...
6
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1answer
347 views

The importance of relations between automorphic forms and arithmetic functions

As I understand things, one of the classical reasons to care about modular forms was their relation to interesting arithmetic functions/counting questions, i.e. on sums of squares and partitions. When ...
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2answers
961 views

A mystery sequence

This question arose from the recent one, roots of a polynomial linked to mock theta function?. Let $$ g(x):=\sum_{k=0}^\infty x^k\prod_{j=1}^{k-1}(1 + x^j)^2\\=1+x+x^2+3 x^3+4 x^4+6 x^5+10 x^6+15 x^7+...
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181 views

Representation of a number as a sum of co-prime numbers

Conventions We will call the following product the canonical expansion of $a$: $$a = \prod\limits_{p_i\in\mathbb{P}}p_i^{\alpha_i}.$$ If $a$ and $b$ are co-prime we write $$a \perp b.$$ $\beth$-...
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83 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)_{\...
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2answers
254 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 ...
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0answers
139 views

Mersenne number with small Carmichael function

Let $\lambda(\cdot)$ be the Carmichael function. I'm trying to understand the magnitude of the smallest values of $\lambda(2^n - 1)$, when $n$ runs over the positive integers. Precisely my question is:...
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2answers
906 views

Does 53 diverge to infinity in this Collatz-like sequence?

This function has been explored a bit at MSE (in June 2016): \begin{eqnarray} f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\ f(n) &=& \lfloor n/4 \rfloor \; \textrm{otherwise} \...
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1answer
341 views

Does anyone recognize this exponential sum?

For $a$, $b$ two integers, let $(a,b)$ denotes their gcd. We define the following exponential sum : $$G_q(n):=\sum_{d|q,~(d,q/d)=1}{e^{2i\pi n\frac{dd'}{q}}}$$ for $n$ a non-negative integer and $q$ ...
4
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0answers
238 views

The properties of Pos

Given $n\in\mathbb{N}$, and $f:\mathbb{N}^*\rightarrow \mathbb{N}$, let define $Pos$ as: $$Pos(f)(n)= |\{x \leq n, f(x)=f(n)\}|$$ When given $n\in\mathbb{N}$, this function gives the 'position' of $...
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242 views

On even almost perfect numbers other than powers of two

(Note: This question is an improved version of and has been cross-posted from this MSE post.) Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) = 2x - 1$, then we call $x$ an ...
8
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2answers
552 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: ...
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2answers
474 views

Are there multiplicative functions which are not rational?

Vaidyanathaswamy calls an arithmetic function rational if it is the convolution of some finite collection of functions which are either completely multiplicative or inverse to a completely ...