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

0
votes
0answers
14 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, ...
3
votes
2answers
110 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,...
0
votes
0answers
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?
2
votes
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
vote
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 $...
3
votes
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/...
0
votes
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
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
16 views
-3
votes
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
vote
1answer
122 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
vote
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
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 ...
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 $...
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 ...
1
vote
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,...
6
votes
0answers
173 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
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
vote
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 ...
0
votes
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
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
votes
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
votes
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 $...
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 ...
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{...
19
votes
0answers
214 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
vote
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
vote
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
votes
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
votes
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
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 ...
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}...
0
votes
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
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 $\...
4
votes
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 ...
0
votes
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 \\ &...
1
vote
0answers
255 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 ...
3
votes
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}\...
0
votes
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 ...
25
votes
1answer
846 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 \\ +1&+1&-2&+1&+1&-2&...
0
votes
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 ...
0
votes
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
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 $...
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 ...
0
votes
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 ...
2
votes
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
votes
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\...