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13
votes
1answer
578 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 ...
2
votes
0answers
50 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) ...
1
vote
1answer
123 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 ...
2
votes
0answers
128 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
vote
1answer
153 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 ...
0
votes
1answer
162 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 ...
1
vote
0answers
39 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 ...
-2
votes
1answer
59 views

Simple equation to distribute points in a game [closed]

I need to create a equation to distribute points for users in following game: There are x users that play a game. If only one of them hit he gets max points. If all of them hit each gets min points. ...
4
votes
0answers
216 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 ...
5
votes
0answers
257 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 ...
-1
votes
1answer
183 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 > ...
1
vote
2answers
175 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 ...
3
votes
1answer
326 views

Menon’s identity

I also put this question in stackexchange, but remained unanswered. http://math.stackexchange.com/questions/506996/menons-identity Let $G$ be a group of order $n$. Consider an action of $U_n$, the ...
2
votes
1answer
246 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 + ...
4
votes
0answers
180 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 $\Delta(n)\le d(n)$ ...
2
votes
1answer
564 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 ...
1
vote
5answers
2k 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
7
votes
3answers
542 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 ...
15
votes
2answers
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 ...
3
votes
0answers
258 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 $k$ ...
1
vote
2answers
437 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 ...
16
votes
3answers
608 views

Extending arithmetic functions to groups

Thinking along the lines of Tom Leinster's fascinating recent question, I'm wondering more generally about how to extend questions about natural numbers to groups, with the cyclic groups representing ...
12
votes
2answers
2k views

Generalized Euler phi function

Let $n$ be an integer, there is a well-known formula for $\varphi(n)$ where $\varphi$ is the Euler phi function. Essentially, $\varphi(n)$ gives the number of invertible elements in ...
1
vote
2answers
723 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 ...