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

learn more… | top users | synonyms

20
votes
0answers
226 views

Sums of the form $\sum_{d|n} x^d$

Let $$S(x,n) = \sum_{d|n} x^d, n \in \Bbb N$$ Do these sums appear in the literature? What are they called if they do and what is known about them?
10
votes
0answers
165 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
0answers
157 views

Is the Euler function $\phi$ constant in arbitrarily large intervals?

Is it true that for every $k \in \mathbb{N}$ there exists a natural number $x$ such that $\phi(x)=\phi(x+1)=\cdots=\phi(x+k)$, where $\phi$ is the Euler's totient function? I thought about this ...
6
votes
0answers
174 views

Are known these identities, that I've deduce using Mobius inversion formula?

I would to know if this formula is right and know (these formula are the same by exponentiation), since I deduce this easily by a standar way (perhaps there are mistakes) using Mobius inversion from $$...
5
votes
0answers
180 views

Summation of a function 2

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}\cdots p_k^{c_k}$ where ...
5
votes
0answers
138 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
0answers
146 views

Arithmetic Derivative

In Calculus, whenever we see a constant and want to take the derivative of it, it always is $0$. However in Number Theory, we have something called the arithmetic derivative in which we can ...
4
votes
0answers
332 views

LCM of binomial coefficients and related functions

I know about the following identity: $$\displaystyle \text{lcm} \left( {n \choose 0}, {n \choose 1}, ... {n \choose n} \right) = \frac{\text{lcm}(1, 2, ... n+1)}{n+1}$$ 1) Is there any method to find $...
4
votes
0answers
78 views

Derivatives, discrete and continuous, of $(1/\sqrt{n})\cos (t\log n)$ and $(1/\sqrt{n})\sin (t\log n)$ and Cauchy-Riemann equations

For any arithmetical function $f(n)$, we define its derivative to be $f'(n)=f(n)\cdot \log n$ for $n\geq 1$ (see for example [1], page 45 or Wikipedia). Fact. The functions $u(n,t)=(1/\sqrt{n})\...
4
votes
0answers
208 views

To prove $\phi(mn)\phi(d)=\phi(m)\phi(n)d$ without explicitly computing the phi function values

If $m,n$ are positive integers with g.c.d.$(m,n)=d$ , then we can show by explicitly computing respective totients that $\phi(mn)\phi(d)=\phi(m)\phi(n)d$, I want to know, is there any more elegant way ...
4
votes
0answers
369 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
0answers
70 views

how to show $\lim \sup_{n \to \infty} \frac{\phi(n+1)}{\phi(n)}= \infty$ and $\lim \inf_{n \to \infty} \frac{\phi(n+1)}{\phi(n)}= 0$?

Let $\phi (n)$ be the Euler's totient function . Thenhow to prove that the set $\{\dfrac{\phi(n+1)}{\phi(n)}: n \in \mathbb Z^+\}$ is unbounded above that is how to show $\lim \sup_{n \to \infty} \...
3
votes
0answers
119 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 ...
3
votes
0answers
86 views

Approximate how the Numbers $n$ such that Mertens' function is zero grow.

Is it possible to approximate how the "Numbers $n$ such that Mertens' function is zero" grow?
2
votes
0answers
31 views

Any formulæ for products of consecutive cyclotomic polynomials?

I am interested in any information about coefficients of the polynomials $\Psi_n(x):=\prod_{k=1}^n\Phi_k(x)$, where $\Phi_k$ are the cyclotomic polynomials. I realize of course that this is a ...
2
votes
0answers
22 views

Characters appearing naturally in arithmetic functions

Let $r_2(n)$ denote the number of representations of $n$ as a sum of 2 squares. It is well-known that $$r_2(n) = 4\sum_{d \mid n} \chi(d),$$ where $\chi$ is the non-principal character modulo $4$. ...
2
votes
0answers
80 views

A challenge question in elementary number theory!

Find an expression for the following sum: $$\sum_{i:(i,n)=1}(i-1,n)$$ I guess that this sum equals to $\phi(n)d(n).$
2
votes
0answers
154 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 ...
2
votes
0answers
74 views

Relation between Dirichlet convolution and Bell series and convolution of functions and the Fourier transform?

We define the Dirichlet convolution of two arithmetic functions $a,b:\mathbb{N}\to\mathbb{C}$ to be $$ (a*b)(n)=\sum_{d\mid n}a(d)b\left(\frac{n}{d}\right). $$ Given a prime $p$, we define the Bell ...
2
votes
0answers
87 views

Smallest prime p with N a quadratic residue mod p

Let $N$ be a square-free number equal to 2 or 3 $(\mod 4)$. Let $P(N)$ be the first odd prime, not a factor of $N$, for which $N$ is a quadratic residue. On average, $N$ would be a non-residue for ...
1
vote
0answers
14 views

Does make sense define a gauge for the integral $\int_2^x\frac{\sum_{n\leq t}\Lambda(n)}{t}dt$, where $\Lambda(n)$ is the von Mangoldt funtion?

I try encourage to me to study and understand the definition of gauge integral. See for example this reference Schechter, The Gauge integral where is explained the definition with an example. It is ...
1
vote
0answers
38 views

On relationships between the general terms of sequences from different equivalences to the Riemann Hypothesis

The following are simple deductions using easy calculations for inequalities and limits. I define the following sequences, whose shape is inspired in Nicolas, Robin and Lagarias, respectively, ...
1
vote
0answers
18 views

Sum over square divisors is multiplicative proof verification

I would like someone to verify my proof of the following claim, which I have been using to solve some problems about proving series identities in Ch. 11 of Apostol's analytic number theory text. Let $...
1
vote
0answers
64 views

Pos properties.

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 $...
1
vote
0answers
30 views

What's about $ \sum_{n=1}^{\infty} \frac{ \mu\left( \sigma (n)\right)}{n^3} ,$ where $\mu(n)$ is Möbius function and $\sigma(n)=\sum_{d\mid n}d$?

Let $ \mu (n)$ the Möbius function and $ \sigma (n)$ the sum of divisors function, then the arithmetical function $g(n)= \frac{ \mu\left( \sigma (n)\right)}{n^3} $ isn't multiplicative since $gcd(2,...
1
vote
0answers
16 views

On a criterion for almost perfect numbers using the abundancy index

Let $\sigma(x)$ denote the sum of the divisors of $x$, and let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. A number $y$ is said to be almost perfect if $\sigma(y) = 2y - 1$. In a preprint ...
1
vote
0answers
25 views

Can we capture the definition to be a Mersenne prime in an identity only involving the arithmetic function $S(n)=\sum_{k=1}^n\text{nmod k}$?

Let $S(n)=\sum_{k=1}^n\text{nmod k}$ the sum of remainder function, denoting $\sigma(n)$ as the sum of divisors function, it is know that for each $n>1$ $$S(n)-S(n-1)=2n-1-\sigma(n),$$ is as ...
1
vote
0answers
23 views

Can you discuss when $\sum_{k=1}^n\sum_{m=0}^{k-1}\sec\left(\frac{2\pi m n}{k}\right)$ is defined and when is an integer?

It is know that when we use the trigonometric addition formula for the tangent $$\tan(\alpha+\beta)=\frac{\tan(\alpha)+\tan(\beta)}{1-\tan(\alpha)\tan(\beta)},$$ taking $\alpha=\beta=x$ then from $\...
1
vote
0answers
45 views

Sketch of a possible equivalence with Riemann hypothesis

From Robin's equivalence (see [1]) and the following trigonometrics identitites, I ask to me if it is feasible write vagues equivalences using this strong result and if these equivalences will be ...
1
vote
0answers
37 views

Numbers $n$ whose prime factors are $2$ and $5$ if and only if $\sum_{k=1}^{rad(n)}\mu(k)k=0$?

In the literature if defined the arithmetical function $rad(n)$ as $1$ if $n=1$ and for $n>1$ by $rad(n)=\prod_{p|n}p$. It is obviously a multiplicative function. Too we know the Mobius function $\...
1
vote
0answers
45 views

What happens to the $\color{red}s$ in Möbius' Inversion Formula?

At the end of the Wiki page on Möbius' Inversion Formula, the following relation is given: $$ g(x) = \sum_{m=1}^\infty \frac{f(mx)}{m^\color{red}s}\quad\mbox{ for all } x\ge 1\quad\...
1
vote
0answers
57 views

Modified arithmetical functions from a modified Möbius function

In this post when $n>1$ we assume that $n=\prod_{k=1}^{\omega(n)}p_{n}^{e_{k}}$ its factorization in prime powers, where $\omega (n)$ is the number of distinct primes. It is well known the ...
1
vote
0answers
70 views

What's is the name of this function?

A function, $f:\mathbb{N}\to\mathbb{N}$, is defined in the following way, \begin{equation} f(n)=\#\{m\mid m\leq n\text{ and there does not exists any integer }m'>m\text{ such that }m\text{ divides }...
1
vote
0answers
83 views

Bezout Coefficient for polynomials in sage?

I want to find the bezout coefficient for those 2 polynomials : $f = 1+x-x^2-x^4+x^5$ and $g = -1+x^2+x^3-x^6$ when I use the gcd function in sage the output is : ...
1
vote
0answers
131 views

Summing the reciprocal of $\phi(n)^2$

I am currently reading Vaughan's book 'The Hardy-Littlewood method' and am working on exercise 3.3 (page 37). I have tried working through a special case but that didn't help either. More concretely, ...
1
vote
0answers
37 views

Number of triples $(a, b, c)$ with $1 \leq a,b,c \leq n$ which are coprime ($gcd(a,b,c)=1$)

Number of ordered triples $(a, b, c)$ with $gcd(a, b, c) = 1$ and $1 \leq a, b, c \leq n$ can be computed using the following formula: $$ C(n) = \sum_{k=1}^n\mu(k) \left \lfloor \frac{n}{k} \right \...
1
vote
0answers
30 views

gcd of product of exponents of prime factors and product of prime factors

Let $n = \prod\limits_i p_i^{k_i}$. I want to express $$ \gcd(\prod\limits_i k_i, \prod\limits_i p_i) $$ as an arithmetic function (i.e get rid of gcd). Is that possible? Thanks!
1
vote
0answers
56 views

All all hypo-multiplicative functions linear combinations of quasi-multiplicative functions?

A function $f(n)$ is multiplicative if, for all coprime $m$ and $n$, $f(m)f(n)=f(mn).$ It is called quasi-multiplicative by many authors if $f(m)f(n)=f(1)f(mn)$, and the two coincide when $f(1)=1.$ (...
1
vote
0answers
110 views

Multiple Dirichlet convolutions

I have been playing around with Dirichlet convolutions. As a reminder, take two arithmetic functions $f,g$, then their Dirichlet convolution is defined as the arithmetic function with: $(f\star g)(n) ...
1
vote
0answers
74 views

Summation of no. of divisors

Let $d(n)$ = no. of divisors of $n$ and $(d(n))^2$ = square of no. of divisors of n. Let $$S(N) = \sum_{n=1}^{N} d(n)$$ and $$S_2(N) = \sum_{n=1}^{N}(d(n))^2$$ $$S(N) = 2\sum_{n=1}^{\left\lfloor\sqrt{...
1
vote
0answers
262 views

Can an odd perfect number be divisible by $101$?

Preamble - This question is an offshoot from the following earlier questions here at MSE: Can an odd perfect number be divisible by 825? Can an odd perfect number be divisible by 165? Odd perfect ...
1
vote
0answers
132 views

Generalizing a result on sums involving Euler's function

Motivation: It's known that there is a constant $0<K$ such that for any natural number $N$, $KN\leq \frac{\varphi(1)}{1}+\frac{\varphi(2)}{2}+\cdots+\frac{\varphi(N)}{N}$ (with $\varphi$ being ...
0
votes
0answers
31 views

Can do you repeat these calculations combining the explicit formula and Nicolas criterion, on assumption of the Riemann Hypothesis?

I did easy calculations to get for $x=N_k=\prod_{n=1}^k p_k$ the kth primorial, combining the so-called explicit formula$\dagger$ for the second Chebyshev function and Nicolas criterion for the ...
0
votes
0answers
15 views

Upper bounds for the modulus of $f(s)=\prod_{n=1}^\infty \left( 1-\frac{\sigma(n)}{n^3}s\right)$

Let the complex variable $s=x+iy$, and $$\sigma(n)=\sum_{d\mid n}d$$ the sum of divisors function (is a known multiplicative function in number theory, for example $\sigma(1)=1$ and $\sigma(6)=1+2+3+6=...
0
votes
0answers
22 views

Get an upper bound of $\left| F(1+it) \right|$ in an example of Perron type formula

From Proposition 3 of Tao, A cheap version of Halasz’s inequality, I know how get for example upper bounds for $x,T\geq 1$ $$\frac{1}{x}\sum_{n\leq x}\frac{\mu(n)\log n}{n}\ll\int_{-T}^{T} \left| \...
0
votes
0answers
18 views

Is there a quickest algorithm than Atkin's sieve for primes detection with <O(N) complexity?

Le code C adaptable aux langages de hauts niveaux est exporté en listing ci-après : ...
0
votes
0answers
13 views

conditional in basic arithmetic

So I am creating a CSS Framework (for web), which aim to be very responsive. So here's my problem: Is there a way to get/compute with only using +, -, * and / (basic arithmetic) to replace the if ...
0
votes
0answers
13 views

Associativity of the Dirichlet Convolution Product

How can you prove that the convolution product of aritmetical functions is associative, and that it is distributive in respect to the addition? The book that i'm reading states that (F_a, ) is a ...
0
votes
0answers
21 views

On the density of solitary numbers

Let $\sigma(x)$ denote the sum of the divisors of $x$, and let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. If $X$ is the unique solution of $$I(X) = \dfrac{a}{b}$$ (for a given rational ...
0
votes
0answers
20 views

It is possible an application of Shapiro's Tauberian theorem for $\sum_{n\leq x}\frac{|M(n)|^{1+\alpha}}{\pi(n)^{1+\beta}}\left[\frac{x}{n}\right]$?

I would like to know if I can find some $\alpha,\beta\geq 0$ such that defining the sequence $a(1)=1$ and for $n>1$ as $$a(n)=\frac{|M(n)|^{1+\alpha}}{\pi(n)^{1+\beta}},$$ where $M(n)=\sum_{k\leq n}...