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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References: rigorous algorithms for elementary computations in base-b with complexity estimates
Definitions/Notation: Fix positive integers $b$ and $M$. Consider the set of real numbers which can be exactly expressed with $2M+1$ coefficients in base $b$, defined by
$$\mathcal{X}(b,M):=\{x\in \...
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Identities to go from $\sum_{n\leq x} \mu(n) \log \frac{x}{n}$ to $M(x) = \sum_{n\leq x} \mu(n)$?
Let $\mu$ be the Möbius function. Say we have a bound on $\check{M}(x) = \sum_{n\leq x} \mu(n) \log \frac{x}{n}$ of the form $|\check{M}(x)|\leq \epsilon x$ for all $x\geq x_0$.
It is then easy to ...
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First occurrence of formula for $\sum_{n\leq x} \mu(n) \log n$ in terms of $\psi(y)-\lfloor y\rfloor$?
The identity contained in the last two displayed equations in the following passage (from page 110 in Ayoub's An Introduction to the Analytic Theory of Numbers, 1963) gives us right away a simple ...
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Brun-Titchmarsh for sum over square divisors
Let $f(n)$ be a nonnegative arithmetic function satisfying
$f(p^l) \leq A_1^l$ for all primes $p$, integers $l\geq 1$, and some constant $A_1$;
$f(n) \leq A_2 n^\varepsilon$ for all $\varepsilon > ...
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Is it true that $\left\{\frac{\sigma(n)}{\varphi(n)}:\ n\in\mathbb{Z}_{\geq 1}\right\}=\{r\in\mathbb Q:\ r\ge1\}$?
For any positive integer $n$, let $\sigma(n)$ be the sum of all positive divisors of $n$.
Clearly, $\sigma(n)\ge n\ge \varphi(n)$ for all $n\in\mathbb{Z}_{\geq 1}$, where $\varphi$ is Euler's totient ...
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Square roots and prime numbers
Definitions:
Here I present a novel conjecture using basic mathematical tools like the sum of the
divisors of an integer $n$ called $\sigma(n)$, the sum of the squares of the positive divisors of n ...
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Behavior of $m(x)\sqrt{x}$ where $m(x)=\sum_{n\leq x}\frac{\mu(n)}{n}$
Let $M(x) = \sum_{n\leq x} \mu(n)$ and $m(x) = \sum_{n\leq x} \frac{\mu(n)}{n}$, where $\mu(n)$ is the Möbius function.
We know that (it is not the best known bounds):
$$\limsup_{x \to \infty} M(x)x^{-...
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Where have you encountered the following arithmetic function?
The following arithmetic function is studied by Zagier in connection with values at odd negative integers of zeta functions of real quadratic fields:
$$e_r (n)=\sum_{\underset{|x|\leq n}{x^{2}\equiv n\...
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Possible implications of the bound $\sum_{n\leq x}\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)=O\left(x\right)$
Let $\lambda(n)$ be the Liouville function and consider the Goldbach-type problem $\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)$. Assume the Riemann hypothesis, the ...
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A question on Euler's totient function
With reference to the Euler's totient function $\phi(\cdot)$, given any $n \in \mathbb{Z}^+$, it's quite straightforward to find $\phi(n)$.
In contrast, given $n \in \mathbb{Z}^+$, even though there ...
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Do all nonnegative integers appear in A051521?
For every positive integer $n$, $\tau(n)$ is the number of divisors of $n$. If we list the ratio of each positive integer $n$ to $\tau(n)$，they form a rational sequence
1,1,3/2,4/3,5/2,3/2,…
Because $\...
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Relationship between two types of partition functions
Referring to this unanswered question on MS, I'm posting the same question here:
For $s\in \mathbb{C},\Re(s)>1 $, consider:
$$\prod_{k=1}^{\infty}\prod_{n=2}^{\infty}\frac{1}{1-n^{-ks}}= \prod_{k=1}...
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Approximation of partial sum over prime omega function
I asked the question in Math StackExchange. Link: https://math.stackexchange.com/questions/4765476/approximation-of-partial-sum-over-prime-omega-function
I haven't got any response yet. Here are the ...
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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 $h$,
$$d(n+h)...
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Are there infinite numbers of the form $\sigma_1(n)=\sigma_1(m)=p$, or is there only one?
I put forward a hypothesis in number theory, it is as follows.$ \sigma_1(n)=\sigma_1(m)=p$, where $\sigma_1$ is the divisor sum function, $n,m\in \mathbb N$, and $p$ is prime. I recently noticed and ...
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Estimating a sum involving the von Mangoldt function
I'd like to know the estimate of the following sum
$$\sum_{n\leq x}\sum_{d|n}\Lambda(d)\frac{\phi(d)}{d} $$
where $\Lambda(d)$ is the von mangoldt function and $\phi(d)$ is the Euler totient function. ...
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On Carmichael function and aliquot parts of odd perfect numbers
I've asked nine months ago this question on Mathematics Stack Exchange with identifier 4430381 and same title. There is not answer for this question on Mathematics Stack Exchange, I wondered if this ...
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Around similar inequalities than an inequality due to Nicolas, that involve products of consecutive Ramanujan primes
This is cross-posted (and this post is a version to ask just around the veracity of Conjecture 1) as the post with identifier 3594907 and same title), that I've edited on Mathematics Stack Exchange ...
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On equations with arithmetic functions [closed]
Is this good topic for research:
equations with arithmetic functions, for example equations like $\varphi(n)=\sigma(n)$ or $\varphi(n)+\sigma(n)=d(n)$ ?
If Anyone here have an advise please tell me ...
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Arithmetic billiards, prime numbers and the Goldbach conjecture
I've edited the following post on Mathematics Stack Exchange, (now closed, at that date I'm suspended) with identifier 4510963, please let me to know if you've some doubt or I can improve the post.
On ...
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Upper bound on minimum number of prime factors in short intervals
Suppose that $H = H(X)$ is some quantity growing with $X$. Are there any bounds on $$F(X, H) = \min_{X < n\le X + H} \omega(n)?$$
It isn't hard to obtain a lower bound $\max_{x\sim X} F(X, H)\gg \...
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A similar inequality for the Dedekind psi function, than an inequality stated by Schinzel
I would like to ask about the next question that seems to me interesting. I know an article that was written by Andrzej Schinzel in which he stated Lemma 2. In this post we denote the Dedekind psi ...
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On the behaviour for the quotient involving Fermat numbers of $\frac{\psi(F_m)}{F_m}$ where $\psi(x)$ denotes the Dedekind psi function
In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
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A conjecture concerning the equation $\sigma\left(\square\right)=\text{prime}$
I can deduce the following simple proposition, the definitions for $\sigma(x)$ the sum of divisors functions and $\varphi(x)$ the Euler totient function are assumed. After I present a conjecture that ...
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Periodic sequences of integers generated by $a_{n+1}=\frac{\operatorname{rad}(pa_{n})}{p}+\frac{\operatorname{rad}(qa_{n-1})}{q}$
Let's define the radical of the positive integer $n$ as
$$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\ p\text{ prime}}}p$$
and consider the sequence
$$a_{n+1}=\frac{\operatorname{rad}(p\cdot a_{n})...
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A definition related to pseudoprimes and the Dedekind psi function
In this post we consider that $\psi(k)$ denotes the Dedekind psi function. Wikipedia has an artcle dedicated to this arithmetic function Dedekind psi function defined for a positive integers $m>1$ ...
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On the density of a particular subset of integers
Given a positive integer $n$ in the standard form
$$n=\prod_k p_k^{\alpha_k}$$
and the arithmetic function (investigated by Erdős in this paper)
$$A(n)=\sum_k \alpha_k p_k$$
let's define the subset $E$...
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For which $n$ is $\sum_{k=1}^n 1 / \varphi(k)$ an integer?
For which positive integers $n$ is the sum $\sum_{k=1}^n 1 / \varphi(k)$ an integer? Here $\varphi$ is the Euler totient function.
The question is a "totient-analog" of the well-known result ...
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Number of different factorizations
I define $\nu(n)$ the number of different factorizations for an integer $n$. I know there are papers about $\delta(n)$ the number of dividers for an integer $n$ (Landau, Euler, Dirichlet) but I still ...
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Help with R. Ryan's "A simpler dense proof regarding the abundancy Index."
I'm reading Richard Ryan's article "A simpler dense proof regarding the abundancy index" and got stuck in his proof for Theorem 2. The Theorem is stated as follows:
Suppose we have a ...
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How to estimate sums over arithmetic progressions?
For $x>1$
$$
N(x)=\sum_{0<n<x \\n \equiv 1 \pmod 4\\ n\text{ squarefree}} 1
$$
How to estimate $N(x)$'s order? (Like $N(x) \sim Ax$)
Furthermore, for $n=p_1p_2\cdots p_v$, define $\alpha (n)=...
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Arithmetic properties of positively reduced $2\times 2$-matrices
Call a $2\times 2$ matrix with coefficients in $\{0,1,2,3,\ldots\}$
positively reduced if any row or column reduction (given by replacing a row/column by itself minus the other row/column) produces
at ...
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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
$$
\sum_{n\leq x} \...
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Correlating the von Mangoldt function with periodic sequences
The Dirichlet inverse of the Euler totient function is:
$$\varphi^{-1}(n) = \sum_{d \mid n} \mu(d)d \tag{1}$$
and the von Mangoldt function can be expanded/computed as:
$$\Lambda(n) = \sum\limits_{k=1}...
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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 ...
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Have any proposals been advanced for the analytic continuation of the divisor function?
While I was working on the evaluation of a certain series, the following limit came up:
\begin{align} \lim_{n \to 1} \frac{d(n)-1}{n(n-1)} &= \lim_{n \to 1} \frac{d'(n)}{2n-1} \\
&= d'(1) .\...
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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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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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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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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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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
$$\...
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Sign changes of a sequence
Let $f$ be an arithmetical function. Suppose that $f(n)>0$ if $n$ is in an integer set $A$ and that $f(n)<0$ for another integer set $B.$ Is there a result from number theory or an elementary ...
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Where can I find the problem by Lagarias?
Jeffrey Lagarias proved, unconditionally, that:
$$
\sigma(n)<H_n+2\exp(H_n)\log(H_n)\qquad n>1
$$
This was posed as a problem in:
J. C. Lagarias, Problem 10949: A generous bound for divisor ...
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$\frac{\sigma(n)}{n} < e \ln \ln (n)$ is true?
In Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187–213 (pdf)
we find the following result:
If the Riemann hypothesis is true ...
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Arithmetical function comparable to sine function [closed]
I was wondering if there exists or can we construct (using known arithmetic functions) an arithmetical function that has the same behaviour of the function sine or comparable to it (I mean that ...
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Generalization of the The Liouville Lambda function
Let $n=p^{\alpha_1}_1 \cdots p^{\alpha_m}_m,$ and define
$$\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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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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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 ...
