Questions tagged [arithmetic-functions]
For questions on arithmetic functions, i.e. real or complex valued functions defined on the set of natural numbers.
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When is $\varphi(n)$ one less than $\omega(n)$ [duplicate]
When is $$\varphi(n) = \omega(n)+1$$
for $$1<n<1000$$
where $\varphi(n)$ represents numbers not exceeding $n$ coprime to $n$, and $\omega(n)$ represents numbers not exceeding $n$ that do not ...
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Does the following GCD divisibility constraint imply that $\sigma(m^2)/p^k \mid m$, if $p^k m^2$ is an odd perfect number with special prime $p$?
The topic of odd perfect numbers likely needs no introduction.
In what follows, denote the classical sum of divisors of the positive integer $x$ by $$\sigma(x)=\sigma_1(x).$$
Let $p^k m^2$ be an odd ...
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Improving $I(m^2)/I(m) < 2^{\log(13/12)/\log(13/9)}$ where $p^k m^2$ is an odd perfect number with special prime $p$
In what follows, let $I(x)=\sigma(x)/x$ denote the abundancy index of the positive integer $x$, where $\sigma(x)=\sigma_1(x)$ is the classical sum of divisors of $x$.
The following is an attempt to ...
- 8,010
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Proof for formula of sum of factors of $n$
For a given composite number $n=a^pb^qc^r \cdots$. Now the number of factors it has are ${(p+1)(q+1)(r+1)...}$ and the sum of all its factors are ${(a^0+a^1+...+a^p)(b^0+b^1+...+b^q)(c^0+c^1+...+c^r).....
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Applying convex multiplicative functions to Brocard's Problem
Brocard's problem asks if there are integer solutions to $n! = (x-1)(x+1)$ other than the cases of $n =$ $4$, $5$, $7$. Knowing the only shared divisor of the factors on the right is 2, would it be ...
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Contiguous generalized hypergeometric functions modifying only denominator variables
The standard contiguous relations for the Gaussian Hypergeometric Functions
can be stacked/repeated to relate $_2F_1(a,b;c;z)$ in many ways
to sums of the formats $_2F_1(a\pm,b\pm ; c\pm ;z)$ , see e....
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How may I show $\sum_{d \mid n} \frac{\mu(d)^2}{\phi(d)} = \frac{n}{\phi(n)}$? [duplicate]
I wish to show the identity
$$\sum_{d \mid n} \frac{\mu(d)^2}{\phi(d)} = \frac{n}{\phi(n)},$$
where $\mu$ is the Möbius function defined by
$$\mu(n) = \begin{cases}(-1)^k & \text{$n=p_1 \dots p_k$,...
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How to reduce division with uprounding to integer-arithmetic operations with downrounding if the denominator is not necessarily an integer?
Let 𝑖∈ℕ₀ be an unknown variable and 𝑐∈ℚ₊ be a known constant such that 𝑐≥2 and 100𝑐 ∈ ℕ₊. (So we can pre-compute anything regarding 𝑐, e.g., represent 100𝑐 as a product of powers of primes.) We ...
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Number of primitive Dirichlet characters of certain order and of bounded conductor
Writing $q(\chi)$ for the conductor of a Dirichlet character $\chi$, one can show using Mobius inversion that
$$\#\{\text{$\chi$ primitive Dirichlet characters}\,:\,q(\chi)\leq Q\}\sim cQ^2.$$
My ...
user1296122
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Important Subgroups of Arithmetical Functions [closed]
I am taking a course in Analytic Number Theory. The main object of study is arithmetical functions. Moreover, if we look at the arithmetical functions which do not vanish at $1$, then they form a ...
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How to make arithmetic function continuous?
Suppose that we have an arithmetic function $f(x)$ defined as follows:
What are the methods in the literature that will make this function continuous and differentiable? However, it should be noted ...
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Expressing Numbers Without Any Decimal Presumptions
I have long been uncomfortable with how numbers in alternative bases are expressed. Alternative bases are marketed as transcending our arbitrary base-$10$ conventions, but I wonder if they really ...
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How to take derivative of an arithmetic function?
Arithmetic functions are defined from natural numbers to complex numbers. Therefore, they are not continuous in the analytic sense and consequently cannot be differentiated analytically. However, we ...
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Book reference for studying Dirichlet Convolution
Now I am studying elementary number theory, I am interested in arithmetic function, I have studied Burton's Number Theory but I can't find Dirichlet Convolution as a particular topic, I will be highly ...
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Relationship between two types of partition functions
After downvoting my previous thread, here is a more detailed explanation of my question. For $s\in \mathbb{C},\Re(s)>1 $, consider:
$$\prod_{k=1}^{\infty}\prod_{n=2}^{\infty}\frac{1}{1-n^{-ks}}= \...
