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

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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 ...
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\...
-1
votes
1answer
36 views

Counting divisors of a number

Let m be any positive integer and consider $\Sigma_{d|m} \frac{1}{d} $. I wish to ask whether there is a closed form expression for the above sum.
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 ...
-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
55 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
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 ...
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
vote
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 ...
1
vote
2answers
39 views

With $s(n)=\sum_{k=1}^n n \bmod k$, can be justified that $\forall\epsilon>0$ let us $\lim_{n\to\infty}\frac{s(n-1)}{\epsilon+s(n)}=1?$

Denoting as $$s(n)=\sum_{k=1}^n n \bmod k$$ the sum of remainders function (each remainder is defined as in the euclidean division of integers $n\geq 1$ and $k$). See [1] for example. For examples $...
10
votes
3answers
359 views

Arithmetical Functions Sum, $\sum_{d|n}\sigma(d)\phi(\frac{n}{d})$ and $\sum_{d|n}\tau(d)\phi(\frac{n}{d})$

$$\sum_{d|n}\sigma(d)\phi\left(\frac{n}{d}\right)=n\tau(n) ,\\ \sum_{d|n}\tau(d)\phi\left(\frac{n}{d}\right)=\sigma(n)$$ The problem (7.4.15) of Burton's Elementary Number Theory has been request to ...
1
vote
1answer
45 views

Using Dirichlet convolution where f = μ ∗ μ (Mobius) to find f(24)?

I am confused about the Dirichlet convolution and how it is used. Does it take two entirely different arithmetic functions? And knowing that f = μ ∗ μ (the Mobius function), why does the question I ...
2
votes
1answer
74 views

Infinite sum of a function $g(n)=\sum_{d|n \; d\ne n}g(d)$

Let the function $g:\Bbb{N}\to\Bbb{N}$ be defined as $$g(n)=\sum_{d|n \; d\ne n}g(d)$$ with $g(1)=1$, how can we evaluate a sum like $$\sum_{i=0}^\infty{g(15^i)\over15^i} \tag1$$ Need we find a ...
0
votes
0answers
45 views

Show that if $n$ is even then $\sum_{d|n}\mu(d)\phi(d)=0$ using only Dirichlet Convolution propertys (without multiplicative function concepts)

Show that $\displaystyle\sum_{d|n}\mu(d)\phi(d)=0$ using only Dirichlet Convolution propertys (without multiplicative function concepts). I suspect you have to use that $1\ast \mu=I$ and $f\ast 1=id$...
4
votes
2answers
137 views

Was this arithmetic Möbius/Mangoldt function ever used for something?

Definition: Let $n=\prod_k p_k^{c_k}$, with $p_k \in \mathbb P$ and $$ A(n)=\sum_{d|n} \mu(d)\Lambda(d)=\sum_\limits{c_k\neq 0} \log p_k , $$ with the Möbius function $\mu(n)$, which is: $...
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 $\...
2
votes
2answers
28 views

On $\sigma(n)<n+\frac{n}{2}\log\frac{n+1}{n-1}+\frac{n}{4} \left(\sum_{\substack{d|n,1<d<n}}\log\frac{(n+d)(n+1)}{(n-d)(n-1)}\right)$

I've derived for $n>1$ and $\sigma(n)$ the sum of divisor function $\sum_{d|n}d$ the following inequality $$\sigma(n)<n+\frac{n}{2}\log\frac{n+1}{n-1}+\frac{n}{4} \left(\sum_{\substack{d|n,1<...
3
votes
1answer
39 views

The sum $\sum_{n\leq x}\sum_{\substack{1\leq k\leq n \\ gdc(k,n)=1}}cos^2\pi \frac{k}{n}$ diverges as $x$, when $x$ tends to infitity

I want to know if it is possible find an easy proof (this is without an use of an strong result) of Question. Prove that the following sum diverges as $x\to\infty$ $$\sum_{n\leq x}\sum_{\...
0
votes
0answers
32 views

On $\sum_{k\nmid n}k$, where the sum is over the integers $1\leq k\leq n$ such that $k\nmid n$, and perfect numbers

If we define the arithmetic function $\delta(n)$ as the sum of integers $1\leq k\leq n$ such that $k\nmid n$, we have by Gauss statement $\sum_{k=1}^n k=n(n+1)/2$, that $$\sigma(n)+\delta(n)=\frac{n(...
0
votes
2answers
58 views

Formula with $\sigma(n)$ and $\phi(n)$ is not an integer if $n$ is not square-free

The question Let $n$ be an integer, let $\sigma(n)$ be the sum-of-divisors function, and let $\phi(n)$ be Euler's totient function. Prove that $\frac{\phi(n)\sigma(n) + 1}{n}$ is an integer if $n$ ...
0
votes
0answers
45 views

Partial GCD - Sum

$\sum\limits_{n = 2}^{M} \sum\limits_{m = 1}^{R} GCD(m,n) $ $R = (\lfloor\dfrac{N}{n}\rfloor-n) \% n $ $M = \lfloor\sqrt{N}\rfloor $ I calculated that - For N=10 the sum is 1 For N = 100 the ...
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 ...
2
votes
1answer
34 views

On Abel summation $\sum_{e<n\leq x}\left(\mu(n)\cdot\int_2^n\frac{ds}{\log s}\right)$, where $\mu(n)$ is the Möbius function

By Abel's identity for $Li (x)=\int_2^x\frac{ds}{\log s}$, $a(n)=\mu(n)$ the Möbius function and $[y=e,x]$ (see Theorem 4.2, page 77 of [1]) and an application of Fundamental Calculus Theorem we ...
0
votes
1answer
33 views

Number of zeros of a polynomial modulo n is a multiplicative function

Let $f$ be a polynomial with integer coeffcients. For $n\geq1$ let $N_f(n)$ denote the number of pairwise incongruent solutions of $f(x)=0$ mod n. I need help proving that $f$ is a multiplicative ...
4
votes
0answers
334 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 $...
3
votes
1answer
77 views

Looking an asymptotic for $\sum_{k\leq x}\Lambda(k)e^k$, where $\Lambda (n)$ is the von Mangoldt function

Using Abel's identity (see Theorem 4.2 in page 77 of [1]) and Prime Number Theorem (Theorem 4.4 in page 75) I compute $$\frac{1}{x}\sum_{k\leq x}\Lambda(k)e^k\sim 1\cdot e^x-\frac{1}{x}\int_1^x \psi(...
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})\...
0
votes
0answers
28 views

On $\sum_{d|n}(\mu(n/d)\sigma_0(d)/2)\log d$, where $\sigma_0(n)$ and $\mu(n)$ are, respectively, the number of divisors and Möbius functions

Let $f(n)=\sum_{d\mid n}\log d$. Then using Möbius inversion formula (see Wikipedia) and $\prod_{d\mid n}d=n^{\sigma_{0}/2}$ (see [1], page 47) where $\sigma_{0}(n)$ is the number of positive divisors ...
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\...
19
votes
2answers
201 views

For all $n$ there exists $x$ such that $\varphi(x)<\varphi(x+1)<\ldots<\varphi(x+n)$

Let $\varphi$ be the Euler's function, i.e. $\varphi(n)$ stands for the number of integers $m \in \{1,\ldots,n\}$ such that $\text{gcd}(m,n)=1$. Let $n\ge 2$ be a positive integer. Show that there ...
6
votes
2answers
221 views

Arithmetic Derivatives: Arithmetic Logarithmic Derivative Problem

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
1answer
119 views

Infinite exponentiation $n^{n^{n^{…^n}}} \equiv m \pmod q$ , find m?

let $(n,q) \in \mathbb N^{*^2}$ I was wondering if it was possible to find a function $f_q$ such that : $f_q(n)=m$ where $m$ is such that $n^{n^{...^n}} \equiv m \mod q$ or at least an easy way to ...
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 $4$. ...
-1
votes
2answers
66 views

Formula to round decimal values

I'm using an application, which offers a feature of creating user-defined functions. Available set of methematical operations which could be incorporated is rather small, namely I can use: addition ...
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 ...
0
votes
1answer
30 views

Composition of analytic function with arithmetic function

Consider an arithmetic function $g$ with codomain $\{a,b\}$ and a function $f$ which is analytic on some domain including $\{a,b\}$. We therefore have $$f(g(n))=\sum_{k=0}^\infty c_k (g(n)-a)^k$$ and ...
3
votes
3answers
318 views

Ramanujan's tau function identity

While studying Ramanujan's tau function, I observed that the function satisfies a beautiful identity that I had not seen previously in the literature. Let $\tau(n)$ be Ramanujan's tau function, such ...
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 ...
1
vote
1answer
45 views

For all $n\in\Bbb N$ does there exist $n_1$ and $n_2$ such that $d(n_1) +d(n_2)=n$

I am trying to show that for each positive integer, $n$, we can find integers $n_1,n_2$ that satisfy $$d(n_1) +d(n_2)=n$$ where $d(n)$ is the divisor function. I am frustrated because even with easy ...
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 }...
0
votes
1answer
31 views

If an arithmetic function has a strong logarithmic mean value, then it has an ordinary mean value.

Say that an arithmetic function f has a strong logarithmic mean value A, and write L(f) = A, if f satis es an estimate of the form \begin{align} \sum\limits_{n=1}^{x}\frac{f(n)}{n} = Alog(x) + B + o(...
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 : ...
2
votes
1answer
46 views

for which values of the pair of integers $(n,k)$ is $p(n,k) =1+\frac{2^{k}-1}n$ is prime?

let $p(n,k)= 1+\frac{2^{k}-1}{n}$ for a positive integer $n,k$ -for which values of the pair of integers $(n,k)$ : $p(n,k)$ is prime ? Any help is very welcom .Thank you
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 ...
3
votes
0answers
87 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?
4
votes
0answers
370 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, ...
4
votes
1answer
50 views

Number of Divisors of most numbers

In the book A Comprehensive Course in Number Theory by Alan Baker. The author mentions that even though the average order of $\tau(n)$ is $\log n$, almost all numbers have about $(\log n)^{\log 2}$ ...
3
votes
1answer
120 views

Convolution identity involving the Möbius function $\sum_{d|n,d>0} |\mu(d)| = 2^{\omega(n)}$

I'm learning about the Möbius Inversion Formula but I'm stuck on an exercise which involves the Möbius function. Let $n\in\mathbb{Z}$ with $n>0$ and let $\omega(n)$ denote the number of distinct ...