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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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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....
R. J. Mathar's user avatar
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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 MΓΆbius function defined by $$\mu(n) = \begin{cases}(-1)^k & \text{$n=p_1 \dots p_k$,...
Robin's user avatar
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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 π‘–βˆˆβ„•β‚€ be an unknown variable and π‘βˆˆβ„šβ‚Š be a known constant such that 𝑐β‰₯2 and 100𝑐 ∈ β„•β‚Š. (So we can pre-compute anything regarding 𝑐, e.g., represent 100𝑐 as a product of powers of primes.) We ...
AlMa1r's user avatar
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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 ...
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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 ...
ALNS's user avatar
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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 ...
user10478's user avatar
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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 ...
Severus' Constant's user avatar
3 votes
2 answers
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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 ...
Albert's user avatar
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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}}= \...
Mohammad Al Jamal's user avatar
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$\sum_{k = 1}^{\infty} k\lfloor\frac{n}{k} \rfloor = 1 + \sum_{k = 1}^n \sigma_1(n)$

For any $f: \Bbb{N} \to \Bbb{Z}$ there exists a unique transformed function $F:\Bbb{N} \to \Bbb{Z}$ such that: $$ f(n) = \sum_{k = 1}^{\infty}F_k\lfloor\frac{n}{k}\rfloor $$ For example, set $F_1 = f(...
Daniel Donnelly's user avatar
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Approximation of partial sum over prime omega function

The prime omega function $\omega(n)$ counts the number of distinct prime factors of a natural number $n$, and can be defined as $\omega(n)=\sum_{p \mid n}1$. Let $S(N)=\sum_{n=1}^{N}n\omega(n)$. Let $\...
piepie's user avatar
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How fast does the proportion guaranteed by dirichlet converge?

I'm working on a counting problem and I'm using Dirichlets theorem (weak form) at some point in the counting. The problem is I don't know how fast something converges and I'm not very knowledgeable in ...
Bruno Andrades's user avatar
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Connection between multiplication table $n * k$ and partial sums of the partial sums of the Dirichlet inverse of the Euler totient function.

I am watching this video: L-functions and the Langlands program (RH Saga S1E2) This reminds me of a recurrence: Let $c=1$ and a recurrence be: ...
Mats Granvik's user avatar
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Verify if exists $n$ such that $\phi(n)$ equals $\frac{n}{d}$, for a constant $d$

Taking $n$ for $n=p_1^{a_1}...p_k^{a_k}$ (being $p_i$ prime numbers)and applying it to Euler's totient function, we would get $\phi(n) = p_1^{a_1-1}...p_k^{a_k-1}(p_1-1)...(p_k-1)$ If we take $d=2$, ...
FeCostaPa's user avatar
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1 answer
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How to get the Euler product for $\sum_{n = 1}^{\infty} \mu(n)/\phi(n^k)$.

Let $\mu$ be the Mobius function, and $\phi$ Euler's totient function. I am reading a proof found in this paper (Theorem 2 on page 17), and I can't quite figure why I'm getting something different ...
matt stokes's user avatar
5 votes
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If an arithmetic function is multiplicative, non-zero at a prime, and "prime-linear", is it the identity?

Let $f:\mathbb{N}\to\mathbb{N}\cup\{0\}$ be a function. Let $f(1)=1,$ and $f(ab)=f(a)f(b)$ whenever $\gcd(a,b)=1.$ Note that I am assuming that $f$ is multiplicative but not completely multiplicative. ...
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A problem on von Mangoldt function.

Suppose $n$ is odd natural number. Define $r(n)=\sum_{n_1+n_2+n_3=n} \Lambda(n_1)\Lambda(n_2)\Lambda(n_3)$ and $r'(n)=\sum_{p_1+p_2+p_3=n}(\log p_1)(\log p_2)(\log p_3)$ where $p_1,p_2,p_3$ are ...
Subhadip Chowdhury's user avatar
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Does an odd perfect number have a divisor (other than $1$) which must necessarily be almost perfect?

Let $\sigma(x)=\sigma_1(x)$ be the classical sum of divisors of the positive integer $x$. Denote the aliquot sum of $x$ by $s(x)=\sigma(x)-x$ and the deficiency of $x$ by $d(x)=2x-\sigma(x)$. ...
Jose Arnaldo Bebita Dris's user avatar
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1 answer
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Almost exact formula for sum of Euler Phi function $\sum_{k=1}^n \varphi(k)$

Solving a problem, I needed a formula for $\Phi(n)=\sum_{k=1}^n \varphi(k)$, where $\varphi(k)$ is Euler's notorious totient function, that counts the number of numbers between $1$ and $k$ that are ...
Zima's user avatar
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Pointwise Extension of Multiplicativity

A standard exercise in elementary number theory is to prove that if two multiplicative functions agree on the set of prime powers, then they are identical arithmetic functions. Consider the following ...
user02138's user avatar
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Is there an algorithm to solve number manipulation problems?

There are this kind of problems where, given a certain amount of the same numbers, it's needed to manipulate them with functions or operators in a way to get a certain result. It's allowed to glue ...
Rusurano's user avatar
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Is this proof for the divisibility constraint $\sigma(q^k)/2 \mid n$ correct, where $q^k n^2$ is an odd perfect number with special prime $q$?

Let $N = q^k n^2$ be an odd perfect number given in Eulerian form, i.e. $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Define the GCDs $$G = \gcd(\sigma(q^k),\...
Jose Arnaldo Bebita Dris's user avatar
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Arithmetic Progression/Sequence | Does it include terms prior to the initial term?

I’m trying to find the right term to describe points along a linear equation f(x) = mx + b, but where the domain is restricted to the set of integers. EDIT: Also the slope and y-intercept are integers....
Ryan Pierce Williams's user avatar
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Is this disproof for the Descartes-Frenicle-Sorli Conjecture that $k=1$, if $p^k m^2$ is an odd perfect number, valid?

Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. It is known that $$D(p^k)D(m^2)=2s(p^k)s(m^2) \tag{0}$$ where $D(x)=2x-\...
Jose Arnaldo Bebita Dris's user avatar
4 votes
1 answer
71 views

Weird arithmetics with prime numbers...

Hello here is a problem with which I struggle quite a bit: For any integer $n \geq 2$, with prime factor decomposition $n = p_1^{Ξ±_1}\times p_2^{Ξ±_2}\times ... p_k^{a_k}$, let $f(n) = Ξ±_1^{p_1}\times ...
DeltaXY's user avatar
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Is the set of divisor indicator functions at $x^2 - 1$: $B' = \{(d \mid x^2 - 1) : d \in \Bbb{N}\}$ a $\Bbb{Z}$-linearly independent set?

We know that the set $B = \{(d \mid x) = \begin{cases} 1, d \mid x \\ 0, \text{ otherwise} \end{cases}, \ d \in \Bbb{N}\}$ forms a $\Bbb{Z}$-module Schauder basis for the module $M =\{ \Bbb{N} \to \...
Daniel Donnelly's user avatar
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2 answers
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Is there an analytical solution to the inequality $\frac{1 + k({p}^{(k+1)/2)})}{p^k} < \frac{p}{p - 1}$ if one were to bound $k$ in terms of $p$?

My question is as is in the title: Is there an analytical solution to the inequality $$\frac{1 + k({p}^{(k+1)/2)})}{p^k} < \frac{p}{p - 1}$$ if one were to bound $k$ in terms of $p$? Here, $p \...
Jose Arnaldo Bebita Dris's user avatar
1 vote
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136 views

Why is it that, if there are no odd perfect numbers, then there are no other $3$-perfect numbers, apart from the six known, as of the year $1643$?

Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $x$. A number $N$ is said to be $k$-perfect if $\sigma(N)=kN$ where $k$ is a positive integer. The number $1$ ...
Jose Arnaldo Bebita Dris's user avatar
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1 answer
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What is the Dirichlet serie of The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$?

The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$ and defined by $$ A(n)=\sum \limits_{p^{\alpha}\parallel n}\alpha p $$ is this serie calculated ...
Es-said En-naoui's user avatar
1 vote
0 answers
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Characterization of Möbius-monotonicity

We say that an arithmetic function $f:\mathbb{Z}^+\to\mathbb{C}$ is MΓΆbius-monotone if $\forall n\geq1:(\mu*f)(n)\geq0$ (where $*$ denotes de Dirichlet convolution), i.e. if there exists a non-...
K. Makabre's user avatar
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1 answer
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On the prime factorization of $n$ and the quantity $J = \frac{n}{\gcd(n,\sigma(q^k)/2)}$, where $q^k n^2$ is an odd perfect number

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\...
Jose Arnaldo Bebita Dris's user avatar
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1 answer
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On a consequence of $G \mid I \iff \gcd(G, I) = G$ (Re: Odd Perfect Numbers and GCDs)

Let $N = q^k n^2$ be an odd perfect number given in the so-called Eulerian form, where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
130 views

Show the function for which the Dirichlet generating series is $\zeta(2s)$ using only $\tau,\varphi,\sigma\text{ and }\mu$ or some explicit formula.

I'm trying to find the function with Dirichlet generating series $\zeta(2s)$, I know that this relates somehow to the Liouville function but I am trying to express it in terms of only the standard ...
awgya's user avatar
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3 votes
1 answer
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Generating function for the "two variable totient"?

Recall that Euler's totient function $\varphi(n)$ is defined as the number of natural numbers $<n$ which are coprime to $n$. The Dirichlet series generating function for $\varphi(n)$ is $$ \sum_{n\...
3 votes
1 answer
324 views

What's so special about the number 24 in the definition of the Ramanujan tau function?

I'm learning about the Ramanujan tau function $\tau \colon \mathbb{N} \to \mathbb{Z}$ defined by $$\tau(1)q + \tau(2)q^2 + \tau(3)q^3 + \ldots = q \prod_{n = 1}^{\infty} \left(1 - q^n\right)^{24}.$$ ...
my2cents's user avatar
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Product of all reduced residues in relation with the function $\mu$: $\prod_{1\leq a\leq n,(a,n)=1}a=n^{\varphi(n)}\prod_{d|n}(d!/d^d)^{\mu(n/d)}$

We know that for any arithmetic function $f$ the Mobius inversion formula gives its inversion. Hence $$ F(n)=\prod_{d|n}f(d)\implies f(n)=\prod_{d|n} F(n/d)^{\mu(d)}.$$ The above statement can be ...
barbatos233's user avatar
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1 answer
195 views

Can this inequality involving the deficiency and sum of aliquot divisors be improved? - Part II

This MSE question (from April 2020) asked whether the inequality $$\frac{D(n^2)}{s(n^2)} < \frac{D(n)}{s(n)}$$ could be improved, where $D(x)=2x-\sigma(x)$ is the deficiency of the positive integer ...
Jose Arnaldo Bebita Dris's user avatar
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1 answer
97 views

Which arithmetic functions satisfy $\sum_{d \mid n} f(\frac{n}{d}) k^d \equiv 0 \pmod{n}$ for all positive integers $n,k$?

There are $$\frac{1}{n} \sum_{d \mid n} \varphi\left(\frac{n}{d}\right) \cdot k^d$$ different $k$-ary necklaces of length $n$ as a result of PΓ³lya's enumeration theorem, where $\varphi$ is Euler's ...
Fikilis's user avatar
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1 vote
1 answer
131 views

On the equation $s(n^2) = \left(\frac{q-1}{2}\right)\cdot{D(n^2)}$, if $q^k n^2$ is an odd perfect number with special prime $q$

Let $N$ be an odd perfect number given in the so-called Eulerian form $$N = q^k n^2$$ where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. In what follows, let ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
95 views

On the second moment of prime divisor function

Let $\omega(n)$ denote the number of distinct prime divisors of $n$. I learned from Cojocaru & Murty that $$ \sum_{n\le x}\omega(n)^2=x(\log\log x)^2+O(x\log\log x).\tag1 $$ I wonder whether it is ...
TravorLZH's user avatar
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0 votes
1 answer
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If $2^r b^2$ is an even almost perfect number that is NOT a power of two, does it follow that $r=1$?

(The following are taken from this preprint by Antalan and Dris.) Antalan and Tagle showed that an even almost perfect number $n \neq 2^t$ must necessarily have the form $2^r b^2$ where $r \geq 1$, $\...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
70 views

Does $\sigma(n^2)/q \mid q^k n^2$ imply $\sigma(n^2)/q \mid n^2$, if $q^k n^2$ is an odd perfect number with special prime $q$? - Part II

(Preamble: This inquiry is an offshoot of this MSE question.) Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ as $I(x)=\...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
98 views

Has this strange real-valued binary function been researched?

Let $X \neq \emptyset$ be an arbitrary set. Let $$ d : X \times X \to \mathbb{R} $$ be a real-valued function with the following properties: (1) $\forall x, \in X: d(x,x) = 0,$ (2) $\forall x,y \in X:...
Joker123's user avatar
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0 votes
1 answer
67 views

Are these valid proofs for the equation $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$, if $q^k n^2$ is an odd perfect number with special prime $q$?

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$. It is known that $$i(q)=\gcd(n^2,\sigma(n^2))=\frac{\sigma(n^2)}{q^k}=\frac{n^2}{\sigma(q^k)/...
Jose Arnaldo Bebita Dris's user avatar
2 votes
1 answer
47 views

On a curious PDE involving $I(q^k)+\frac{2}{I(q^k)}$

Let $I(x)=\sigma(x)/x$ be the abundancy index of the positive integer $x$, where $\sigma(x)=\sigma_1(x)$ is the classical sum of divisors of $x$. Consider $$f(q,k)=I(q^k)+\frac{2}{I(q^k)}$$ where $5 \...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
117 views

Are the (Bezout) coefficients for $\gcd(n^2, \sigma(n^2)) = q\sigma(n^2) - 2(q - 1)n^2$ (where $q^k n^2$ is an odd perfect number) unique?

In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the deficiency of $x$ by $D(x)=2x-\sigma(x)$, and the aliquot sum of $x$ by $s(x)=\sigma(x)...
Jose Arnaldo Bebita Dris's user avatar
5 votes
1 answer
400 views

Why counting the number of 1 digits that appear in all integers in 0-9, 0-99, 0-999, 0-9999 follow an arithemtic-geometric sequence?

I noticed that 0-9 = has only 1 '1' 0-99 = has 20 '1's [1,10,11,12,13,14,15,16,17,18,19,21,31,41,51,61,71,81,91] 0-999 = 300 0-9999 = 4000 It follows the formula of n = number of digits in the ...
Zamar's user avatar
  • 161
-6 votes
1 answer
641 views

If $p$ and $2p + 1$ are odd primes and $n = 4p$, prove that $Ο†(n + 2) = Ο†(n) + 2$. [closed]

Prove: if $p$ and $2p + 1$ are odd primes and $n = 4p$, show that $Ο†(n + 2) = Ο†(n) + 2$. I'm stuck on this simple question about Euler's theorem. Any help would be welcome. Thanks in advance!
user avatar
0 votes
1 answer
87 views

Is the argument used in this proof that $k=1$ logically sound, where $q^k n^2$ is an odd perfect number with special prime $q$?

The topic of odd perfect numbers likely needs no introduction. Euler proved that a hypothetical odd perfect number $N$, if one exists, must have the so-called Eulerian form $N=q^k n^2$, where $q$ is ...
Jose Arnaldo Bebita Dris's user avatar

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