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 views

Does existence of mean of arithmetic function imply it is bounded except on a set of density zero? [on hold]

Let $f(n)$ be an arithmetic function whose mean is finite. Is $f$ bounded outside a set of density zero?
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0answers
18 views

Computational experiments around the divisor functions, and the matrix $A= \begin{pmatrix} n & \sigma(n) \\ 1 & \sigma_0(n) \end{pmatrix}$

Let for integers $n\geq 1$ the sequence of matrices defined by $$A= \begin{pmatrix} n & \sigma(n) \\ 1 & \sigma_0(n) \end{pmatrix},$$ where $ \sigma(n)=\sum_{d\mid n}d$ is the sum of divisors ...
2
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3answers
56 views

The arithmetic function $\lambda(n)=(-1)^{a_1+\cdots +a_k}$

Define $\lambda(1)=1$, and if $n=p_1^{a_1}\cdots p_k^{a_k}$, define $$\lambda(n)=(-1)^{a_1+\cdots +a_k}$$ How can I see that $$\sum_{d\mid n}\lambda(d)=\begin{cases} 1 \,\,\text{ if $n$ is a square}\\...
1
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0answers
17 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 $...
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1answer
28 views

Question about multiplicative arithmetic functions

Let $f,g:\Bbb N\to \Bbb C$ be multiplicative arithmetic functions, i.e. $$\gcd(m,n)=1\implies f(mn)=f(m)f(n)$$ and same for $g$. We can also assume $f(1)\neq 0$ and $g(1)\neq 0$ if necessary. How can ...
0
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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=...
3
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2answers
46 views

Estimate for $\sum_{q=1}^{M}\frac{\varphi(q)}{q^{2}}$ Related to Bourgain Paper [duplicate]

Let $N\gg 1$ be a large parameter, which I ultimately want to let tend to infinity. I am reading an old paper of Bourgain, where he claims the lower bound (Equation 2.50, pg. 118) $$\sum_{q=1}^{N^{1/...
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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| \...
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0answers
16 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 : ...
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2answers
63 views

If $\dfrac{1}{\infty}=0$ then I can prove that $0 = 1$ [closed]

Given, $\dfrac{1}{\infty}=0$, then $1=0 \cdot \infty = 0$ (because $0$ times any number or values is $0$ and here that number is infinity). Which gives us $1=0$ i.e, $0=1$. Hence proved....
1
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1answer
43 views

Compute the Dirichlet inverse of $f(n)=\frac{1}{1+|\mu(n)|}$, where $\mu(n)$ is the Möbius function

Let for integers $n\geq 1$ the arithmetical function defined by $$f(n)=\frac{1}{1+|\mu(n)|},$$ where $\mu(n)$ is the Möbius function. Note that $f(1)=\frac{1}{2}\neq 0$, and $f(n)$ isn't ...
0
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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 ...
1
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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 $...
0
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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 ...
1
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0answers
29 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
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1answer
26 views

On the Density of Deficient Odd Numbers and Abundant Integers

Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) < 2x$, then $x$ is said to be deficient, while if $\sigma(x) > 2x$, $x$ is said to be abundant. (Of course, when $\sigma(x) ...
1
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0answers
13 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 ...
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3answers
27 views

Prove that $\bigg\lfloor\frac{m}{p^k}\bigg\rfloor-\bigg\lfloor\frac{n}{p^k}\bigg\rfloor-\bigg\lfloor\frac{m-n}{p^k}\bigg\rfloor$ equal to zero or one

Prove that $\bigg\lfloor\frac{m}{p^k}\bigg\rfloor-\bigg\lfloor\frac{n}{p^k}\bigg\rfloor-\bigg\lfloor\frac{m-n}{p^k}\bigg\rfloor$ equal to zero or one for all $k,m,n\in \mathbb N$. where $m\geqslant n$ ...
2
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1answer
26 views

Perfect numbers of the form $12m+1$ and $\sum_{d\mid n}\frac{1}{\phi(d)}$, where $\phi(m)$ is Euler's totient function

If there are no mistakes combining Exercise 9 a) (Chapter 3, page 71) and Exercise (Chapter2, page 47) of Apostol's Introduction to Analytic Number Theory we can prove easily Lemma. If $n$ is a ...
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0answers
16 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
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1answer
31 views

Is the following inequality involving the sum-of-divisors and Euler totient functions true?

First Question Is the following inequality involving the sum-of-divisors $\sigma$ and Euler totient $\phi$ functions true? $$\frac{\sigma(N)}{N} \leq \frac{N}{\phi(N)}$$ Second Question When $...
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2answers
945 views

Is it something new?

$W(n)$ is the function that counts number of distinct prime divisors of $n$. I have been able to prove for any $m$ consecutive integers starting with $1+a$ with the condition $a\leq (m^2-4m)/4$ , ...
0
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3answers
51 views

How to design the max function for integers using only additions and multiplications?

I want to design a function which outputs the maximum value between two integers, something like this $f(x,y) = \begin{cases} 1, & \text{if } x > y, \\ 0, & \text{otherwise}. \end{...
0
votes
1answer
34 views

A computational experiment about identities involving the sum of remainders function

Let $\sigma(m)$ the sum of divisors function and $$S(m)=\sum_{k=1}^m\text{m mod k}$$ the sum of remainders function, then it is know that for integers $m>1$ $$\sigma(m)+S(m)=S(m-1)+2m-1.$$ On the ...
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0answers
24 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 ...
2
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0answers
30 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 ...
3
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1answer
52 views

Can you give an understandable strategy to compute the asymptotic behaviour of $\sum_{2\leq n\leq x}\frac{\Lambda(n)Li(n)}{n}$?

First I am looking a discussion about if the statements that I've deduced will find a good asymptotic formula, what of them you can discard as non useful,or one better than other, and ... second my ...
1
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1answer
46 views

Exploring the Dirichlet series of the sum of remainder function

I wolud like to learn and understand more some basic facts about Dirichlet series, for wich I want explore the following function, that is called the sum of remainders function, A004125 as Sloane's ...
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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}...
0
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1answer
44 views

Vaughan's identity, a didactic example

I know that Vaughan's identity is one of the methods used in analityc number theory, I would like see an example, I say a simple example of application of this theorem for encourage to study the ...
1
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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 $\...
4
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1answer
33 views

What about $\sum_{\substack{2\leq n\leq y,\text{n prime}}}n\log\log n$ when $y=[x]\to\infty$?

For a real $x\geq 2$ and when we take $y= [x]$ its integer part, I am trying to study the asymptotic size or growth of $$\sum_{\substack{2\leq n\leq y,\text{n prime}}}n\log\log n,$$ I believe that ...
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0answers
47 views

Standard name or notation for the “even part” of an integer?

\begin{align} 0 & \mapsto 0 \\ 1 & \mapsto 0 \\[6pt] 2 & \mapsto 2 \\ 3 & \mapsto 2 \\[6pt] 4 & \mapsto 4 \\ 5 & \mapsto 4 \\[6pt] 6 & \mapsto 6 \\ 7 & \mapsto 6 \\ &...
3
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1answer
52 views

Can you prove this identity involving the divisor sum function?

Let $$\sigma(n)=\sum_{d\mid n}d$$ the sum of divisor function. I've deduced, but I don't know how prove directly $$\sigma(n)=\sum_{m=1}^{n}\sum_{k=1}^{m}(-1)^{nk}\cos\left(\frac{\pi n(2-m)k}{m}\...
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0answers
23 views

Can you analyze this identity involving the sum of divisors function and $rad(n)=\prod_{p\mid n}p$?

Let $\sigma(n)$ the sum of divisors function, then by Mobius inversion formula $$\sigma(n)=n-\sum_{\substack{d\mid n,d<n}}\sigma(d)\mu\left(\frac{n}{d}\right),$$ and since this function is ...
0
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1answer
14 views

Two questions about pseudo equidistributed sequences modulo 1

Let $s_n$ a sequence of positive real numbers such that $$\lim_{n\to\infty}\frac{1}{s_n}=0$$ and $$\lim_{n\to\infty}\frac{s_{[nt]}}{s_n}=t,$$ for every real $t\in[0,1]$. See here, page 4. Question ...
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0answers
18 views

$Φ_n$ is Euler group, $n> 2$ is an integer, and $m$ the number of solutions of the equation $x^2 = 1$ in the ring $Z_n$.

Prove $$\prod_{i ∈ Φ_n} i=(-1)^{\frac{m}{2}}$$ Then what becomes this identity if $n$ is a prime number? I know that if $x^2=1$, we pair the number with its inverse modulo $n$ in the product. ...
0
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3answers
26 views

How is countably infinite addition defined

In the axiom of additivity of probability theory, the concept of a countably infinite sum, i. e. the sum of countably infinitely many real numbers, is used. Could someone please tell me how that kind ...
3
votes
3answers
79 views

An identity involving $[\sigma(n)]^2$

For a positive integer $n$, let $\sigma(n)$ denote the sum of the divisors of $n$. For example, $\sigma(1)=1$, $\sigma(2)=3$, $\sigma(4)=7$, etc. I would like to prove the following identity: For ...
1
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2answers
56 views

Prove that $τ(m^n)$ and $n$ are coprime $(m,n ∈ N^+)$

We have that $τ(n)=\sum_{d|n} 1$ is the number of dividers of n. Dividers of $m^n$ are $1,m,m^2,...,m^n$ then we have that $τ(m^n)=n+1$. Is this correct so far? Now we must prove that $τ(m^n)$ and $...
2
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1answer
49 views

What inequalities similar Lagarias' statement are easy to prove?

Let $$H_n=1+\frac{1}{2}+\cdots+\frac{1}{n},$$ the nth harmonic number and $$\sigma(n)=\sum_{d\mid n}d,$$ the sum of divisor function, for example $\sigma(6)=12$. I believe that this could be a nice ...
0
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0answers
31 views

Is there a name for this discrete version of Jensen, specifically when applied to binomial coefficients?

We have $2k$ integers greater than or equal to $j\geq0$ $a_1+a_2+\dots + a_k=n$ and $b_1+b_2+\dots + b_k=n$. If for all $1\leq i\leq k$ we have $|n/k-a_i|\leq|n/k-b_i|$. Then $\sum\limits_{i=1}^n\...
3
votes
1answer
81 views

What about $\lim_{n\to\infty}\frac{\sum_{k=1}^n s_k\mu(k)}{n}$, for the zeros of Dirichlet eta function $s_k=1+\frac{2\pi k}{\log 2}i$ with $k\geq 1$?

Let for integers $k\geq 1$ the corresponding zeros of Dirichlet eta function of the form $$s_k=1+\frac{2\pi k}{\log 2}i,$$ then we can consider the following puzzle, when we multiply previous ...
6
votes
0answers
172 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 $$...
-1
votes
4answers
56 views

How can one show that $f(n)>n$

Given a function $f\colon \Bbb N^\ast\to \Bbb N^\ast$ such that we have $f(f(n))=f(n+1)-f(n)~\forall~n\in\Bbb N^\ast$. Show that: If $f(n+1)-f(n)>1$ then $f(n)>n$. Since $f(n+1)>f(n)+1$ I ...
1
vote
1answer
53 views

Can be justified this formula for $\zeta(2n+1)$

Can be justified for integers $n\geq 1$ that $$\zeta(2n+1)=\prod_{\text{p, prime}}\frac{1-\sigma(p^{2n})^{-1}}{1-p^{-2n}}?$$ Truly I don't know if I am wrong another time, when I use for an integer $...
3
votes
1answer
54 views

Hints to compute if exists $\lim_{n\to\infty}\sum_{k=1}^n\sigma(k^2)/\sum_{k=1}^n\sigma(k)$, which $\sigma(n)=\sum_{d\mid n}d$, and other question

I would like receive hints at least for one of the following problems, these are going from experiments. Can you provide to me hints for at least one of the following problems? I will try put the ...
1
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0answers
44 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 ...
4
votes
1answer
49 views

On a generalization for $\sum_{d|n}rad(d)\phi(\frac{n}{d})$ and related questions

Let $\phi(m)$ Euler's totient function and $rad(m)$ the multiplicative function defined by $rad(1)=1$ and, for integers $m>1$ by $rad(m)=\prod_{p\mid m}p$ the product of distinct primes dividing $...
1
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1answer
41 views

Arithmetic functions & logs

How do you simplify the following equation in terms of other arithmetic functions? $$ f(n)= \sum_{d|n} \mu(d) log(d) ? $$ log(n) is not a multiplicative arithmetic function, so i dont know what to ...