Questions tagged [arithmetic-functions]
An arithmetic function is one whose domain is the positive integers and whose range is a subset of the complex numbers. There are a number of important number-theoretic examples.
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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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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
$$\...
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$\frac{\sigma(n)}{n} < e \ln \ln (n)$ is true?
In Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187–213 (pdf)
we find the following result:
If the Riemann hypothesis is true ...
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Where can I find the problem by Lagarias?
Jeffrey Lagarias proved, unconditionally, that:
$$
\sigma(n)<H_n+2\exp(H_n)\log(H_n)\qquad n>1
$$
This was posed as a problem in:
J. C. Lagarias, Problem 10949: A generous bound for divisor ...
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Sign changes of a sequence
Let $f$ be an arithmetical function. Suppose that $f(n)>0$ if $n$ is in an integer set $A$ and that $f(n)<0$ for another integer set $B.$ Is there a result from number theory or an elementary ...
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Is $T(n)=\sum_{k=1}^{n}\frac{\lambda(k)\Lambda(k)}{k} \geq 0$ and what is the upper bound of $T(x)=\sum_{n\leq x} \lambda(n)\Lambda(n) $?
Let $\Lambda(n)$ denote the von Mangoldt function: $\Lambda(n)=\log p$ when $n=p^e$ is a prime power ($e\ge 1$) and $\Lambda(n)=0$ otherwise.
and $\lambda(n)$ be Liouville Function, , I'm interested ...
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Arithmetical function comparable to sine function [closed]
I was wondering if there exists or can we construct (using known arithmetic functions) an arithmetical function that has the same behaviour of the function sine or comparable to it (I mean that ...
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Generalization of the The Liouville Lambda function
Let $n=p^{\alpha_1}_1 \cdots p^{\alpha_m}_m,$ and define
$$\lambda_k(n)= (-1)^{ [\frac{\Omega(n)}{k} ]},$$
where $\Omega(n)= \alpha_1 + \cdots + \alpha_k,$ and $[\cdot]$ is the floor function.
For $...
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Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?
This question was posted in MSE in early August 2020. It did garner several upvotes, but did not receive any responses. I have therefore cross-posted it here, hoping that it gets answered.
Let $\...
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Arithmetic expansion of harmonic sum
Note: I have modified the initial question as follows:
Let $w_1, w_2, \ldots, w_d$ be positive weights, and $x_1, x_2, \ldots, x_d$ be positive variables. Now, let us consider the following harmonic ...
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Is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound?
Pillai showed in 1929 that the function $A(n)$ giving the number solutions of the equation $\phi(x)=n$ is unbounded in (S. Pillai, On some functions connected with $\varphi(n)$, Bull. Amer. Math. Soc. ...
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On the density map of the abundancy index
Let $σ$ be the sum-of-divisors function. Let $σ(n)/n$ be the abundancy index of $n$. Consider the density map $$f(x) = \lim_{N \to \infty} f_N(x) \ \ \text{ with } \ \ f_N(x) = \frac{1}{N} \#\{ 1 \...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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$\varphi(m+n)\mid n$ 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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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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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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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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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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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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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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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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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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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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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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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.
$$\varphi^{-1}(n) = \...
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Existence of equation about the product of the divisor sum function
Let $\sigma_k(n)$ be the sum of the $k$-th powers of the positive divisors of $n$ and $\mu(n)$ be the Möbius function.
As Arithmetic function - Wikipedia mentioned, there is an equation that $$\...
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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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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 ...
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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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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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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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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 ...
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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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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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