variation of Euler Totient

to calculate the number of integers co-prime to N and less than N we can simply calculate its ETF( Euler's totient function - Wikipedia, the free encyclopedia . )However to calculate the number of integers co-prime to N but less then X where X < N , how can we modify / calculate it ? I know how to calcuate the ETF but can't proceed how to modify it to get the required result.

Thanks

Re: variation of Euler Totient

Hey pranay.

Being co-prime to an integer means the gcd(a,b) = 1.

But in terms of the totient function, this means that totient(a,b) = totient(a)*totient(b) (which is on the first page of the wiki).

So you know how many numbers up to n-1 are co-prime by using totient and you also have a condition for when two numbers are co-prime. Can you modify this to get an expression to "remove" the specific numbers of integers that are co-prime but equal or larger than x?

Re: variation of Euler Totient

pranay,

I'm afraid that I don't see how to use chiro's hint. However, your function is very close to what's known as the *Legendre Totient Function*. It's denoted $\displaystyle \phi(X, N)$ and is defined to be the number of positive integers less than or equal to X that are coprime to N. Your function can be written as $\displaystyle \phi(X + 1, N)$ since you define your function to be the number of integers (positive, I assume) that are less than X and coprime to N. The following formula expresses the Legendre Totient function in terms of the Moebius function $\displaystyle \mu$ and the floor function $\displaystyle \lfloor \ \rfloor$:

$\displaystyle \phi(X, N) = \sum_{d|N} \mu(d) \left\lfloor\frac{X}{d} \right\rfloor$

where the summation runs over the positive divisors of N. I can provide a proof and/or some references. Just ask or let me know if you have any questions..

HTH

Petek

Re: variation of Euler Totient

Thanks a lot Petek, i too got that formula, but could you please explain as to why this works ?

Re: variation of Euler Totient

The formula in my previous post follows from the inclusion-exclusion principle and the observation that the number of positive integers less than or equal to X and divisible by d equals $\displaystyle \left \lfloor \frac{X}{d} \right \rfloor$.

For example, let's compute $\displaystyle \phi(11, 20) $:

The positive divisors of 20 are 1, 2, 4, 5, 10 and 20.

By direct inspection, the integers 1, 3, 5, 7 and 9 are less than or equal to 11 and coprime to 20, so $\displaystyle \phi(11, 20) = 5$.

The total number of positive integers less than or equal to 11 is $\displaystyle \left \lfloor \frac{11}{1} \right \rfloor = 11$.

From this we have to subtract the number of positive integers less than or equal to 11 that are divisible by 2, which is $\displaystyle \left \lfloor \frac{11}{2} \right \rfloor = 5$, and the number of positive integers less than or equal to 11 that are divisible by 5, which is $\displaystyle \left \lfloor \frac{11}{5} \right \rfloor = 2$.

However, we subtracted the number of positive integers less than or equal 11 that are divisible by 10 twice, so we have to add back $\displaystyle \left \lfloor \frac{11}{10} \right \rfloor = 1$.

Therefore,

$\displaystyle \phi(11, 20) = \left \lfloor \frac{11}{1} \right \rfloor - \left \lfloor \frac{11}{2} \right \rfloor - \left \lfloor \frac{11}{5} \right \rfloor + \left \lfloor \frac{11}{10} \right \rfloor = 11 - 5 - 2 + 1 = 5$,

where we used $\displaystyle \mu(2) = \mu(5) = -1 $ and $\displaystyle \mu(10) = 1$.

(Note that $\displaystyle \mu(4) = 0$, so the term with divisor = 4 doesn't actually appear in the formula. The same is true of the divisor 20, since $\displaystyle \mu(20) = 0$.)

Hope you find this explanation useful.