All Questions
Tagged with arithmetic-functions nt.number-theory
108 questions
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 ...
- 23k
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 ...
- 9,114
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
9
votes
4
answers
4k
views
Averages of Euler-phi function and similar
What are the odds two numbers are relatively prime? This is known to be $\frac{6}{\pi^2}$. The proof involves calculating averages of the Euler phi function.
\[ \phi(1) + \phi(2) + \dotsb + \phi(n) \...
- 22.8k
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 ...
- 367
19
votes
3
answers
4k
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 $\mathbb{Z}/n\...
- 337
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 ...
- 13.5k
5
votes
4
answers
2k
views
Good books on arithmetic functions?
As I was studying the Möbius $\mu$ function and Gram series,
I got myself some pretty nice books:
Ribenboim - The New Book of Prime Number Records
Apostol - Introduction to Analytic Number Theory
...
- 731