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

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187 views

How can the Möbius function be applied to a series?

Given a series $p_n(s)=\sum_{k=1}^n a_k $. I'd like to get $\hat{p}_n(s)=\sum_{k=1}^n \mu(k)a_k $. Think of $a_k=k^{-s}$ for example. If you let $n$ go to $\infty$, you'll see the well-known relation ...
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2answers
41 views

Finding root for the segment - found the formula but it doesn't work for some values - wrong formula?

I have the segment, defined as $(x_1, y_1)$, $(x_2, y_2)$. I know that $y_1\ge 0$ and $y_2 < 0$. I want to compute the root point for that segment. I decided to do it that way: ...
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36 views

Demonstration in arithmetic function

I need help to know (in detail) how to prove that the product of two multiplicative arithmetic functions is a multiplicative arithmetic function. $$$$$f(n)$ and $g(n)$ are functions multiplicative, ...
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190 views

Function that counts the number of divisors of a natural number?

Let Function $f(n)$ be formally defined for natural numbers such that it gives number of distinct divisors of the number n (n and 1 included) For example, $f (12)=6$, then what is a quick way to ...
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61 views

Finding formulas for sums

I know that $\sum_{d \mid n} \mu(d) = 0$ whenever $n >1$, and I know that $\sum_{d \mid n} \phi(d) = n$. How can I use this in order to give a formula for $\sum_{d \mid n} \mu(d)\phi(d)$?
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55 views

Farey Sequence Vector Orthogonality Relation Question

Take the Farey sequence $\mathcal{F}_n$ for $n=39$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\biggr(\exp(2\pi i k a_m)\biggr)_m $$ Since Merten's function for $n=39$ ...
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72 views

What proportion of the natural numbers satisfy the following inequalities?

Let $\sigma_1(n)$ be the sum of the divisors of $n \in \mathbb{N}$, and let $$I(n) = \frac{\sigma_1(n)}{n}$$ be the abundancy index of $n$. What proportion of the natural numbers satisfy the ...
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103 views

Attempted exercise using Littlewood's theorem

This was an exercise to try to show we can use Littlewood's theorem$^1$ to prove that $$\lim_{N \to \infty}\frac{1}{N}\sum_{n=1}^{N} \frac{g(n)}{\log p_n} = 1 \hspace{30mm}(1)$$ If $\vartheta(p_k) ...
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1answer
129 views

Littlewood's 1914 proof relating to Skewes' number

From Littlewood's 1914 theorem (paraphrase): I propose to show there are arbitrarily large values of x for which successively $\psi(x) - x < - K\sqrt{x}\log\log\log x \tag{A}$ $ ...
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33 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 ...
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39 views

Sums involving square of Moebius function

I try to estimate the following sum: $$ \sum_{n \leq x}\mu(n)^2 f(n) $$ where $\mu(n)$ is a Moebius function and $f(n)$ is some multiplicative arithmetic function. If I understand it correctly it is ...
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19 views

An inequality involving Möbius function [duplicate]

For any positive integer $n$ show the inequality holds : $$\left|\sum_{i=1}^{n}\frac{\mu(i)}{i}\right|\le 1$$ I tried induction. when $\mu(n+1)=0$ it is trivial. But what if $\mu(n+1)\ne 0$? I am ...
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53 views

Summation of no. of divisors

Let $d(n)$ = no. of divisors of $n$ and $(d(n))^2$ = square of no. of divisors of n. Let $$S(N) = \sum_{n=1}^{N} d(n)$$ and $$S_2(N) = \sum_{n=1}^{N}(d(n))^2$$ $$S(N) = ...
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1answer
34 views

Arithmetic progressions that generate an infinite number of powers of 2?

For an arithmetic progression of the form $a_i = ki, i \in \mathbb{N}$, the question is trivial - if $k$ is a power of 2, then the progression will generate an infinite number of powers of 2, and no ...
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111 views

Generalizing a result on sums involving Euler's function

Motivation: It's known that there is a constant $0<K$ such that for any natural number $N$, $KN\leq \frac{\varphi(1)}{1}+\frac{\varphi(2)}{2}+\cdots+\frac{\varphi(N)}{N}$ (with $\varphi$ being ...
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2answers
20 views

Find the numbers; Arithmetic Progression.

The sum of four consecutive numbers in an A.P is $28$. The product of the second and third numbers exceeds that of the first and last by $18$. Find the numbers. I thought of this: $$S_{4} = ...
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1answer
86 views

Question about the proof of Theorem 2.19 (Page 38) of the book Introduction to Analytic Number Theory by Apostol

At the last line of the proof: $\lambda^{-1}(n)=\mu(n)\lambda(n)=\mu^2(n)=|\mu(n)|$. Why $\mu(n)\lambda(n)=\mu^2(n)$? How to prove this?
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42 views

Find the term of the G.P.

Find the term of the G.P. $1, 1.2, 1.44, 1.728,... $ which is just greater than $100$. Please somebody explain me the question? All i know is $a=1$ and $r=1.2$ Help :( should it be $T_{n} > ...
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81 views

Without using group theory, How to Prove $n|\phi(a^n-1)$, where $\phi$ is Euler's Totient function. [closed]

Let $\phi$ be Euler's Totient funcion, how to prove this property? If possible can we have an elementary proof without leveraging the group theory? $$n|\phi(a^n-1), \forall n,a>1, \gcd(a,n)=1$$ ...
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22 views

Mean of given Arithmetic function

Find the mean of $a, a+d, a+2d, a+3d,\dots,a+nd$ I have no idea what to do in this question but i have tried the following: $$mean\ \bar{x}= \frac{(a)+(a+d)+(a+2d)+(a+3d)+\cdots+(a+nd)}{n+1} $$ ...
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16 views

how to check arithmetic progression of triangle

I solved the problem with the following text: In a rectangle the sides and the diagonal are an arithmetic progression. Calculate the circumference of the rectangle where the longer side is 44 cm ...
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38 views

A more elegant version of this function?

I challenged myself. The goal was to find a function $f$ with two variables $x$ and $y$ real, which results $1$ if $x=y$ and results $0$ if $x ≠ y$. But, the fonction can only use additions, ...
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86 views

What proportion of the positive integers satisfy $I(n) < \frac{2n}{n + 1} \leq I(n^2)$ < 2?

Let $$I(x) = \frac{\sigma(x)}{x}$$ be the abundancy index of the positive integer $x$. Note that $\sigma(x)$ is the classical sum-of-divisors function. For example, $$\sigma(12) = 1 + 2 + 3 + 4 + ...
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29 views

Number of excellent pairs is equal to $\sigma(n)$

Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair of $(a,b)$ of positive integers is called good if $$a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu ...
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26 views

An inequality with $\omega(n)$ [duplicate]

Prove: For any positive integer $k, N$, $$\left(\frac{1}{N}\sum\limits_{n=1}^{N}\left(\omega (n)\right)^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q}$$ Where $\sum\limits_{q\leq ...
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57 views

Does there exist a counterexample to the inequality $I(n) < 2{\left(\frac{n}{n + 1}\right)}^2$, if $n$ is odd and deficient?

If $n$ is odd and deficient, does there exist a counterexample to the inequality $$I(n) < 2{\left(\frac{n}{n + 1}\right)}^2,$$ where $$I(x) = \frac{\sigma(x)}{x}$$ is the abundancy index of ...
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45 views

What proportion of the positive integers satisfy $I(n^2) < (1 + \frac{1}{n})I(n)$, where $I(x)$ is the abundancy index of $x$?

Let $\sigma(x)$ denote the classical sum-of-divisors function, and let $$I(x) = \frac{\sigma(x)}{x}$$ be the abundancy index of the positive integer $x$. My question is this: What proportion of ...
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115 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 ...
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83 views

Converting loop to a closed form expression? [duplicate]

Possible Duplicate: How to convert this loop into a closed form expression? I have the following code in Python ...
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5answers
131 views

Solve for $m\in\mathbb{N}$ the equation $\phi (m)=12$

Solve for $m\in\mathbb{N}$ the equation $\phi (m)=12$ I found (by trial) $m=\{13,21,26,28,36\}$, but do not know if misinterpreted the problem, but actually I suppose I have to find an equation ...
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71 views

What are arguments to $\frac 00 = Undefined$?

Now, I understand that dividing by zero in any case is undefined. However, in math, there are always exceptions. I'm just really curious...what are the different cases for different answers? For most ...