For questions on arithmetic functions, a real or complex valued function $f(n)$ defined on the set of natural numbers.

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21
votes
2answers
598 views

Primes approximated by eigenvalues?

Consider the matrix starting: $$\displaystyle T = -\begin{bmatrix} +1&+1&+1&+1&+1&+1&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ ...
8
votes
1answer
138 views

Comparing average values of an arithmetic function

Suppose $f(n)$ is a positive real-valued arithmetic function such that $$ \frac1n\sum_{k=1}^nf(k)\sim g(n) $$ for $g(x)$ a monotonic increasing function. What can be said about the asymptotic behavior ...
8
votes
1answer
250 views

Proving that $\omega(N)\neq4$ for an odd perfect number $N$ by hand

Let $\omega(n)$ denote the number of distinct prime factors of a positive integer $n$, and let $N$ be an odd perfect number. It is not difficult to show that $\omega(N)\ge3$. In fact, Nocco already ...
8
votes
0answers
125 views

Elementary references on Robinson Arithmetic, Prim. Recursive fns etc.

I'm in the middle of revising my freely available and much-downloaded introductory notes Gödel Without (Too Many) Tears. (They are a sort of cut down version of part of my Gödel book, and I'm ...
7
votes
2answers
71 views

$\varphi(N)>\pi(N)$?

Is it trivial that $\varphi(N)>\pi(N)$ for sufficiently big integers $N$, where $\varphi$ is Euler's totient function and $\pi$ is the prime-counting function? The only exceptions less than ...
7
votes
1answer
166 views

Is there a complex variant of Möbius' function?

When you're dealing with arithmetic functions, you might have come across the classical Möbius' function $$ \mu(n)=\begin{cases} (-1)^{\omega(n)}=(-1)^{\Omega(n)} &\mbox{if }\; \omega(n) = ...
6
votes
4answers
271 views

How to prove $ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$

how to prove: $$ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$$ $\prod_{d|n} d$ is product of all of distinct positive divisor of $n$, $\tau (n)$ is number (count)of all of positive divisor of $n$
5
votes
2answers
182 views

Sum of $n \sigma(n)$

What is known about the asymptotic behavior of $$ -\frac{\pi^2}{18}x^3+\sum_{n\le x}n\sigma(n) ? $$ It seems to be $O(x^{2+\varepsilon})$ but I cannot prove this.
5
votes
1answer
54 views

Lower bound of Euler phi function times sum of divisors

After some work, I got this nice inequality: $$ \frac{n^2}{2} < \phi(n)\cdot \sigma(n) $$ where $\phi(n)$ is Euler's phi function and $\sigma(n)= \sum_{d|n} d$. I know this is true because I'm ...
5
votes
1answer
108 views

How many numbers are products of $p^p$?

Consider the set $\mathcal{S}=\{1,4,16,27,\ldots\}$ of numbers which are products of numbers of the form $p^p$ for $p$ prime. ($\mathcal{S}$ is A072873 in the OEIS.) Note that multiple primes are ...
5
votes
0answers
135 views

Is the application of $\mu$ on $P_x(s)^k$ analogous to the differentiation $\frac{d^k f(\lambda) }{d\lambda^k}\biggr|_{\lambda=0}$?

Let me start with the following on elementary symmetric polynomials: The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity ...
4
votes
2answers
217 views

Ramanujan 1918 paper

Does anybody know where I can find Ramanujan's paper from 1918 titled "On Certain Arithmetical Functions." It is referenced in wikipedia, under the Ramanujan Summation section, but I cannot find a ...
4
votes
2answers
144 views

Limiting value of $\lim \frac{1}{k}\sum_{n=1}^k \frac{p(n+1)-p(n)}{\log p(n)}$

Empirically it seems $$\lim_{k\to \infty} \frac{1}{k}\sum_{n=1}^k \frac{g(n)}{\log p(n)} = 1\tag{1} $$ in which p(n) is the nth prime and g(n) is the prime gap $p(n+1)-p(n).$ Cramer conjectured ...
4
votes
1answer
466 views

On sums involving Euler's totient function

I've been struggling with the following claim without being able to prove it, so your help would be highly appreciated: Let $\varphi(n)$ be Euler's totient function. Show that there is a constant ...
4
votes
3answers
174 views

Prove there are k consecutive non-squarefree integers

So, I've got a question for class that asks me to prove the existence of arbitrarily long runs of consecutive integers where $\mu(n)$ is zero. I've started the proof, but I need a bit of help midway ...
4
votes
2answers
126 views

Which natural number predicates can be defined in Robinson arithmetic?

I'm especially wondering about the order relation, subtraction, division and exponentiation here: $x \leq y \quad \Leftrightarrow \quad \exists u\ y=x+u$ $z= x-y \quad \Leftrightarrow \quad ...
4
votes
1answer
107 views

Decidable & Recursive predicates

Let $C$ be a decidable predicate in the language of arithmetic HA, that is $$ \vdash (\forall \underline x)\: C(x) \vee \neg C(x).$$ $C$ is recursive if there exists a computable characteristic ...
4
votes
3answers
189 views

Question about $T_n = n + (n-1)(n-2)(n-3)(n-4)$

The formula $T_n = n + (n-1)(n-2)(n-3)(n-4)$ will produce an arithmetic sequence for $n < 5$ but not for $n \ge 5$. Explain why. I think it is because if n is less than five the term with ...
4
votes
1answer
144 views

Image of the Euler phi function

I was reading a text about arithmetic functions, which ofcourse mentioned the Euler phi function. I was wondering whether $\phi(n)$ takes on all positive integer values. The answer doesn't seem so ...
4
votes
2answers
73 views

An uncountable well-ordered subordering of asymptotic growth rates?

Define the following relation $\le$ between arithmetic functions $f$ and $g$ (mappings from $\mathbb{N} \rightarrow \mathbb{N}$): $f \le g := \exists n_0, k: \forall n: n \gt n_0 \implies f(n) \lt k ...
4
votes
2answers
88 views

prove that $\phi(xy) =\phi(x)\phi(y)$ for any $x$ and $y$ with $(x, y) = 1$. [duplicate]

Prove that $\phi(xy) = \phi(x) \phi(y)$ for any $x$ and $y$ with $(x, y) = 1$. I understand the concept, and have done several examples proofing this but cannot put it in "proof form" because unless ...
4
votes
1answer
124 views

Euler phi function, number theory

I am trying to find the value of $$\sum_{n=1}^{N}\sum_{d|n}d*\phi(d)$$ Is there a method to evaluate this for large N? $\phi(d)$ is the Euler phi function.
4
votes
2answers
92 views

Prove: $\sum_{k<n, (k,n)=1} k= \frac{1}{2}n \varphi (n)$

Prove: $\sum_{k<n, (k,n)=1}k = \frac{1}{2}n \varphi (n)$ I have had strep throat and missed the lecture discussing properties of the Euler function. Any help in solving this is appreciated. ...
4
votes
1answer
93 views

Asymptotics for almost all $x$

Theorem 2.2 in Shparlinski 2006 says: For all positive integers $n\le x$ except possibly $o(x)$ of them, the bound $$M(x)\ll\frac{x}{\log x}\exp\left((C+o(1))(\log\log\log x)^2\right)$$ holds. ...
4
votes
2answers
252 views

Elements of finite order in the group of arithmetic functions under Dirichlet convolution.

Let $(G, ∗)$ be the group of arithmetic functions $f : N \to C$ that satisfy $f (1)\neq 0$, with group operation given by the Dirichlet product $∗$. The identity function $I$ is the identity element ...
4
votes
2answers
140 views

Estimating $\sum_{p_2 \leq x} (\log p_2)^2$

This was an exercise to use the approach here to estimate the sum $\sum_{p_2 \leq x} \log (p_2)^2,$ in which $p_2$ are numbers containing two prime factors (repetitions allowed). $\pi_2(x)$ is the ...
4
votes
0answers
293 views

Numbers $n$ such that Mertens' function is zero.

OEIS (A028442) lists the Numbers n such that Mertens' function $$ M(n)=\sum_{k=1}^n\mu(k) $$ is zero: 2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, ...
3
votes
1answer
119 views

Prove that the Möbius function is multiplicative

I'm studying algebra, and I came across some questions on multiplicative functions (that should be number theory though?). One is: prove that mobius function is multiplicative. But I've not been given ...
3
votes
4answers
93 views

What's the meaning of this $(m,n) = 1$

I'm reading this pdf http://rutherglen.science.mq.edu.au/wchen/lndpnfolder/dpn01.pdf I understand some of the expression used in this but I don't understand the part $(m,n) = 1$ Is this a cartesian ...
3
votes
3answers
80 views

Prove that for all $n\in\mathbb{N}$, $\frac{s(n)}{d(n)}\geq \sqrt n$

Prove that for all $n\in\mathbb{N}$ $$\frac{s(n)}{d(n)}\geq \sqrt n$$ where $s(n) = \sum_{d|n} d$ and $d(n) = \sum_{d|n} 1$. Being honest, study some time arithmetic functions, and can not ...
3
votes
1answer
81 views

A sum regarding prime factorization

Prime factorization of $n$ is $n = p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$ Let $f(n) = \left((p_1^{a_1}+1)(p_2^{a_2}+1)(p_3^{a_3}+1)\cdots(p_k^{a_k}+1)\right)$ I want to find the value of ...
3
votes
1answer
64 views

Is there a friendly pair $\{a, b\}$ where both $a$ and $b$ are odd?

Let $\sigma(x)$ denote the sum of the divisors of a positive integer $x$. $\{y, z\}$ is said to be a friendly pair if $$I(y) = I(z),$$ where $I(x) = \sigma(x)/x$ is the abundancy index of $x$. As ...
3
votes
2answers
112 views

Order of a function related to divisors

Let $f(n)=\max(\{d(ab):\ a,b\le n\})$ where $d(m)$ is the number of divisors of $m.$ What is the order of $f$? In particular I'm looking for an asymptotic upper bound.
3
votes
1answer
78 views

Properties of Arithmetic Functions

I was recently working on arithmetic functions and using Perron's formula to obtain asymptotic estimates. One observation I made was that the Dirichlet series often can be written in terms of the ...
3
votes
1answer
410 views

Chebyshev's first $\vartheta(x)$ function question

This was an exercise using the first Chebyshev function, $\vartheta(x)= \sum_{p \leq x} \log p.$ The question is simply how to prove (2) below, the rest is my two thoughts on how to proceed. [Edit: ...
3
votes
1answer
89 views

does the surd $\sqrt[p-1]{p}$ for prime $p$ occur in any context?

for any prime $p \in \mathbb{N}$ use the corresponding symbol $q$ to denote the quantity $$q=p^{\frac1{p-1}}$$ and for $n \in \mathbb{N}$ define: $$ Q_n= \prod_{p \le n} q $$ empty products evaluate ...
3
votes
1answer
106 views

Minimal $x$ for which $\phi(k) > n$ for all $k > x$

It's well-known that $$ \liminf_n\frac{\varphi(n)\log\log n}{n}=e^{-\gamma} $$ and there exists an effective version $$ \varphi(n)>\frac {n}{e^\gamma\log\log n+\frac{3}{\log\log n}} $$ valid for ...
3
votes
0answers
71 views

On the indivisibility of odd perfect numbers by small numbers

A good day to everyone! This question is an offshoot of the following MSE posts: Odd perfect number divisors Can an odd perfect number be divisible by $101$? My question is as follows: Is ...
3
votes
1answer
25 views

Two sets with the same geometric and arithmetic means

There are two sets A and B with equal geometric mean and arithmetic mean. Each element of both sets is odd integer greater than 1. A = B ? Order of elements isn't important.
3
votes
0answers
103 views

Does the Fourier matrix $F_n$ represent a (tensor) multiplicative function?

At "Complex Hadamard Matrices", I found that, two Kronecker (tensor) products of Fourier matrices $k_1$ and $k_2$ are equivalent, if and only if $k_2$ can be obtained from $k_1$ by a combination of an ...
2
votes
1answer
95 views

Is the totient function $\varphi$ invertible?

As title, is the totient function $\varphi: \mathbb{N} \to \mathbb{N}$ invertible?
2
votes
4answers
115 views

Finding the GCD of $50!$ and $2^{50}$

I've been trying to figure out how $n!$ and $x^n$ are related (where x is an integer) for most of the morning - I know it must be the key to unlocking this problem. Up to this point I've only used ...
2
votes
2answers
60 views

how to compute the “Hamming total” of a symmetric group?

The Hamming distance between two binary numbers has an obvious analogue for numbers encoded using a base $\gt 2$. A different analogue is the following. Let $\sigma \in S_n$ be a permutation which ...
2
votes
1answer
105 views

Definition of “totient”

I had always taken the term "totient" to be defined by saying that the totient of a positive integer $n$ is the number of positive integers less than $n$ that are coprime to $n$. Thus, for example, ...
2
votes
3answers
50 views

Where did the sum-of-divisors function come from?

Doing a research project on a few number-theoretic functions, and I was curious, where does the sum-of-divisors function come from? Surely someone thought it up and made it a possibility. I'm talking ...
2
votes
1answer
76 views

A closed form for $\sum_{i\cdot j^k=n}(-1)^i$?

$$\alpha_k(n) \stackrel{\text{def.}}{=} \sum_{i\cdot j^k=n}(-1)^i.$$ Does a closed form exist for $\alpha_k(n)$? For low values of $k$: $$\alpha_0(n)=(-1)^n$$ $$\alpha_1(n)=\begin{cases} ...
2
votes
2answers
69 views

Is there a way to show that $d(n)$, which counts the number of divisors of $n$ is non-increasing? [closed]

Is there a way to show that $d(n)$, which counts the number of divisors of $n$ is non-increasing? I'm trying to use the Cauchy condensation test to show that $\sum_{n\ge{2}}\frac{d(n)}{n\log^2n}$ is ...
2
votes
1answer
150 views

Next asymptotic term of the average order of sigma

$$ \sum_{k=1}^n\sigma(k)=\frac{\pi^2}{12}n^2+O(n\log n). $$ Is the next asymptotic term known? That is, is there a monotonic increasing function $f(x)$ such that $$ ...
2
votes
1answer
42 views

A Möbius Identity

I need to prove the identity $$\sum_{k = 1}^n \mu(k)\left[ \frac{n}{k} \right] = 1$$ where $n$ is a natural number, and $[n]$ denotes the floor function. The proof also should not use the Möbius ...
2
votes
2answers
21 views

Proving an identity of the Möbius function and Euler’s totient function product

Could anyone kindly help me to prove that $$ \sum_{d|n} \mu(d) \varphi(d) = 0 $$ for all even integers $ n \geq 2 $, where $ \mu $ is the Möbius function and $ \varphi $ is Euler’s totient function? ...