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

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4 views

Lagarias and Robin theorems versus multiplicative property

If I use for example Robin's theorem, see here in the section Growth of arithmetic functions, or Lagarias equivalence, see (5) here has sense ask us what is the more sharp inequality for ...
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1answer
8 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
19 views

Arithmetic problems with understanding

I'm having problems with understanding qn 22. Part (III) don't really know what are they asking :|
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13 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 ...
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3answers
23 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
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3answers
75 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 ...
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2answers
54 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 ...
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1answer
43 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 ...
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0answers
27 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 ...
3
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1answer
79 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 ...
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46 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 ...
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4answers
54 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 ...
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1answer
48 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
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1answer
50 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 ...
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38 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 ...
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1answer
34 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 ...
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1answer
36 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 ...
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2answers
34 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 ...
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1answer
32 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.
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1answer
34 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
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1answer
70 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 ...
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0answers
36 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 ...
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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
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2answers
27 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} ...
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1answer
36 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 ...
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2answers
44 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 $n$ ...
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0answers
29 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 ...
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0answers
35 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 ...
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0answers
108 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
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1answer
33 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 ...
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1answer
26 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
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0answers
316 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
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1answer
64 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 ...
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0answers
71 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?
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24 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 ...
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72 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 ...
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44 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 ...
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2answers
100 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: ...
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2answers
187 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
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1answer
101 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 ...
19
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2answers
189 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 ...
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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 $4$. ...
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2answers
62 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 ...
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0answers
47 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 ...
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1answer
27 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
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3answers
243 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 ...
2
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1answer
58 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 ...
7
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0answers
145 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
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1answer
44 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
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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 ...