Questions tagged [arithmetic-functions]
An arithmetic function is one whose domain is the positive integers and whose range is a subset of the complex numbers. There are a number of important number-theoretic examples.
131 questions
0
votes
0
answers
202
views
Representation of a number as a sum of co-prime numbers
Conventions
We will call the following product the canonical expansion of $a$:
$$a = \prod\limits_{p_i\in\mathbb{P}}p_i^{\alpha_i}.$$
If $a$ and $b$ are co-prime we write
$$a \perp b.$$
$\beth$-...
- 251
1
vote
0
answers
90
views
An arithmetic function involving arbitrary (but fixed) number of divisors
I need at least basic information about generating functions of the following class of arithmetic functions, grouped by levels $k$.
Fix some $k$ and some family $\varepsilon_*=(\varepsilon_\sigma)_{\...
- 18.4k
5
votes
2
answers
314
views
Congruences for the non-divisors of Euler's $\phi(n)$
If $n$ is composite, then $\phi(n) < n-1$: hence, there is at least one number $d$ which does not divide $\phi(n)$ but divides$(n-1)$. We shall call $d$ the totient divisor of $n$. The purist will ...
CommunityBot
- 1
1
vote
0
answers
256
views
On even almost perfect numbers other than powers of two
(Note: This question is an improved version of and has been cross-posted from this MSE post.)
Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) = 2x - 1$, then we call $x$ an ...
CommunityBot
- 1
5
votes
1
answer
392
views
Does anyone recognize this exponential sum?
For $a$, $b$ two integers, let $(a,b)$ denotes their gcd. We define the following exponential sum :
$$G_q(n):=\sum_{d|q,~(d,q/d)=1}{e^{2i\pi n\frac{dd'}{q}}}$$
for $n$ a non-negative integer and $q$ ...
- 13.1k
4
votes
0
answers
465
views
The properties of Pos
Given $n\in\mathbb{N}$, and $f:\mathbb{N}^*\rightarrow \mathbb{N}$, let define $Pos$ as:
$$Pos(f)(n)= |\{x \leq n, f(x)=f(n)\}|$$
When given $n\in\mathbb{N}$, this function gives the 'position' of $...
- 18.4k
9
votes
2
answers
740
views
Asymptotics of product of Euler's totient function (A001088)?
Conjecture:
\begin{align}
\lim_{n\to \infty } \, \frac{\left(\prod _{k=1}^n \phi (k)\right){}^{1/n}}{n}\sim 0.2059\text{...}
\end{align}
The numerical result from 100000 terms is:
My questions are:
...
- 109k
5
votes
2
answers
556
views
Are there multiplicative functions which are not rational?
Vaidyanathaswamy calls an arithmetic function rational if it is the convolution of some finite collection of functions which are either completely multiplicative or inverse to a completely ...
- 9,114
0
votes
1
answer
475
views
Sum of digits of a power [closed]
Are there any explicit formula for a sum of digits for a power in the given base? A problem to be specific: find a sum of digits for a number $2^{100}$ in the system with a base 5. In the system with ...
- 54.2k
6
votes
1
answer
331
views
A question about $(0,1]$-valued multiplicative functions
Suppose $f:\mathbb{N}\to [0,1]$ is a multiplicative function (i.e. $f(nm)=f(n)f(m)$ whenever $m$ and $n$ are coprime). Suppose $f$ has non-zero mean, which means
$$
\lim_{N\to\infty}\frac{1}{N} \sum_{...
- 43.7k
2
votes
0
answers
98
views
Discrete "difference" equations that involve changes in both shift and scale
A standard use of the Z-transform ($F(z) = \sum_n (f[n] \cdot z^{-n} )$) is to understand the effect of a difference equation on a signal. For instance:
$y[n] = x[n] + y[n-1]$
$Y(z) = X(z) + Y(z) \...
- 4,907
6
votes
0
answers
333
views
Linear combination of multiplicative functions
Carlitz showed necessary and sufficient conditions for an arithmetic function to be a linear combination of two multiplicative functions. He mentions the possibility of generalizing to $k$ ...
- 9,114
13
votes
1
answer
934
views
Is the set of multiplicatively even numbers thick?
A positive integer is multiplicatively even (odd) if, when decomposed into primes, the sum of the exponents is even (odd).
A subset of the integers is thick if it contains arbitrarily long intervals $\...
- 114k
1
vote
1
answer
163
views
estimate an sum
I need estimate the following sum:
$\sum_{d=1}^{n}\frac{\mu(d)}{d}\sum_{k=1}^{\lfloor n/d\rfloor}\frac{1}{k}\frac{q^k}{1-q^{-kd}}$, where $q>1$ and $\mu$ is the Möbius function.
To obtain the ...
- 96.4k
3
votes
0
answers
443
views
Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula?
My question is on Mobius inversion formula convergence/properties when used with infinite sums of function.
Lets consider (on $\mathbb{R}^{+}$):
$$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$
We call $...
- 1,199
1
vote
1
answer
170
views
Existence of arithmetic function satisfying a certain property
I was interested in an arithmetic function satisfying a certain property, I am not sure at the moment if such thing even exists or not. But I was wondering maybe I could get some hint or idea or input ...
- 424
2
votes
1
answer
281
views
sum over primes involving divisor function (variation of the Titchmarsh divisor problem)
This question was also asked on MSE.
Does there exist an asymptotic estimate for the following sum over primes
$$
\sum_{p\leq x} \frac{\tau(p-1)}{p}\;,
$$
where $\tau(n)=\sum_{d|n}1$ is the divisor ...
CommunityBot
- 1
1
vote
0
answers
80
views
All all hypo-multiplicative functions linear combinations of quasi-multiplicative functions?
A function is called quasi-multiplicative by many authors if $f(m)f(n)=f(1)f(mn)$, a slight generalization of multiplicativity. (Basically a multiplicative function times a constant is a quasi-...
CommunityBot
- 1
1
vote
5
answers
5k
views
The Inverse of the Euler Totient Function
How can we calculate the cardinality of the inverse of Totient function of any positive integer n ?
I tried going through this paper, but I couldn't understand the procedure.
Thanks
- 34.3k
4
votes
0
answers
275
views
A closed formula for this arithmetic function
The following function comes up in my research as part of a sufficient condition for capability of $p$-group of class two and prime exponent. Given a nonnegative integer $m$, express $m$ as a ...
- 7,217
4
votes
0
answers
413
views
Maximal order of Hooley's Delta function?
There is a large literature on Hooley's
$$
\Delta(n)=\max_u\sum_{d|n,\ e^u\le d< e^{u+1}}1
$$
giving its normal and average order. What is known of its maximal order?
Clearly $\Delta(n)\le d(n)$ ...
CommunityBot
- 1
0
votes
1
answer
314
views
Are all known $k$-multiperfect numbers (for $k > 2$) not squarefree?
I asked the following question in MSE four ($4$) days ago, but so far nobody has posted an answer.
The gist of the question is as follows:
Are all known $k$-multiperfect numbers (for $k > 2$...
CommunityBot
- 1
15
votes
2
answers
1k
views
Sum of $\sum_{k=1}^nd(k^2)$
There is a literature dealing with
$$
\sum_{k\le x}d(f(k))
$$
where $f$ is an irreducible polynomial and $d(n)$ is the number of divisors of $n$. Erdos 1952 shows that the sum $\asymp x\log x,$ which ...
- 12.3k
7
votes
0
answers
332
views
$n\varphi(n)\equiv 2\pmod{\sigma(n)}$ as a primality test
It is known from Subbarao, "On two congruences for primality" that $n>22$ is a prime iff $$n\sigma(n)\equiv 2\pmod{\varphi(n)},$$ where $\varphi(n)$ is Euler's function and $\sigma(n)$ is sum of ...
CommunityBot
- 1
1
vote
2
answers
1k
views
Sum of digits iterated
Original version.
I believe that it is an elementary question, already discussed somewhere. But I just have no idea of how to start it properly. Take a positive integer $n=n_1$ and compute its sum of ...
1
vote
2
answers
199
views
What proportion of the positive integers satisfy $I(n^2) < (1 + \frac{1}{n})I(n)$, where $I(x)$ is the abundancy index of $x$?
Let $\sigma(x)$ denote the classical sum-of-divisors function, and let
$$I(x) = \frac{\sigma(x)}{x}$$
be the abundancy index of the positive integer $x$.
My question is this: What proportion of ...
CommunityBot
- 1
7
votes
1
answer
1k
views
Menon’s identity
I also put this question in stackexchange, but remained unanswered. https://math.stackexchange.com/questions/506996/menons-identity
Let $G$ be a group of order $n$. Consider an action of $U_n$, the ...
CommunityBot
- 1
2
votes
1
answer
1k
views
A formula combining Euler $\phi$ and $\gcd$
Let us fix a natural number $N>1$ and $a_1, \ldots, a_n$ natural numbers satisfying $0 \leq a_i < N$, with the property that $1+ \sum a_i$ is divisible by $N$. Let $\phi$ be the Euler totient ...
CommunityBot
- 1
7
votes
3
answers
695
views
The digit sum: $s(na)=s(nb)$
Not that I was serious about the following question, but I think it is a must-to-ask as a follow-up to this MO post.
For integer $n\ge0$, let $s(n)$ denote the sum of the digits in the decimal ...
CommunityBot
- 1
2
votes
1
answer
301
views
What proportion of the positive integers satisfy $I(n) < \frac{2n}{n + 1} \leq I(n^2) < 2$?
Let
$$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 + ...
CommunityBot
- 1
2
votes
2
answers
589
views
Which rationals are sum-of-divisor function quotients
Consider the function $\sigma(n)/n$, where $\sigma$ is the usual sum-of-divisors function. I read somewhere that it is unknown what rational numbers are in fact values of this function (or at any ...
