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

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1answer
45 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 $...
25
votes
1answer
874 views

Primes approximated by eigenvalues?

Consider the infinite matrix starting: $$\displaystyle T = -\left( \begin{array}{ccccccc} +1&+1&+1&+1&+1&+1&+1&\cdots \\ +1&-1&+1&-1&+1&-1&+1 \\...
1
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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 ...
0
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0answers
37 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 ...
-1
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2answers
37 views

On a superior limit involving the multiplication formula for the Gamma function and the divisors $d\mid n$ of a positive integer

I did the specialization for the $m's$ in the multiplication formula for the Gamma function, see the identity (4) in page 250 of Apostol, Introduction to Analytic Number Theory Springer (1976) as the ...
1
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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
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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
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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 ...
12
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1answer
195 views

How to prove that $n\sum_{d\mid n}\frac{|\mu(d)|}{d}=\sum_{d^2\mid n}\mu(d)\sigma\left(\frac{n}{d^2}\right)$?

This is problem 11 part b in chapter 3 of Tom M. Apostol's "Introduction to Analytic Number Theory". A variation on Euler's totient function is defined as $$\varphi_1(n) = n \sum_{d \mid n} \frac{|\mu(...
3
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2answers
112 views

Closed-Form Modular Arithmetic

Is there a way to define modulo division (or functions of modular arithmetic in general) as superposition of (elementary?) functions? For example, the multiplication is first introduced as summation,...
2
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3answers
58 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
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 $...
3
votes
2answers
53 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/...
0
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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=...
0
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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
18 views
-3
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2answers
65 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
123 views

sum over primes involving divisor function (variation of the Titchmarsh divisor problem)

Does there exist an asymptotic estimate for the following sum over primes $$ \sum_{p\leq x} \frac{\tau(p-1)}{p}\;, $$ where $\tau(n)=\sum_{d|n}1$ is the divisor function?
1
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1answer
44 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
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,...
6
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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 $$...
1
vote
1answer
28 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) ...
0
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3answers
28 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
votes
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 ...
0
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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
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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 ...
0
votes
3answers
52 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{...
1
vote
2answers
67 views

Evaluate $\sum_{d\mid N}\Lambda(d)$

For a positive integer $n$, define $$\Lambda(n) = \left\{ \begin{array} {ll} \log p & \mbox{if $n = p^r$, $p$ a prime and $r \in \mathbb{N},$ }\\ 0 & \mbox{otherwise.} \end{...
21
votes
0answers
227 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?
1
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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 $a\leq (m^2-4m)/4$ , ...
1
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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 ...
2
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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 ...
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
vote
1answer
48 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 ...
0
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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
votes
1answer
46 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
votes
1answer
34 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 ...
0
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0answers
48 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
votes
1answer
53 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}\...
0
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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 ...
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 ...
0
votes
1answer
16 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 ...
0
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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. ...
1
vote
2answers
57 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 $...
3
votes
3answers
83 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 ...