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Let $\Lambda(x,n)$ be the the number of totatives of $x$ which are less than or equal to $n$, and $\Phi(x)$ be Euler's totient function.

For now assume $x>n$.

Is there a general formula for $\Lambda(x,n)$? Furthermore, has the result stated below been documented elsewhere?

Let $l = gcd(x,n)$, $x'= x/l$ and $n'=n/l$

I have a proof that...

$\Lambda(x,n) = \frac{n'}{x'} \Phi(x) \pm V$, where the variance $ 0 \leq V \leq \frac{x'-n'}{x'n'} \Phi(x)$

Extra

Actually from what I can gather this result can be generalized further as follows:

Let $\alpha_{n,k}$ be the interval $ \lbrace a_{0+nk}, a_{1+nk}, ..., a_{(n-1)+nk} \rbrace$ where;

a) $k$ is some natural number such that we can choose its value to position this interval somewhere on the non-negative integer line,

b) $a_i=i$ for all non-negative integers $i$

c) $n$ is a natural number, that we can chose as the size of this interval. (Notice this interval always starts with a number divisible by $n$.)

And, let $\Lambda_\chi(x,I)$ be the number of totatives of $x$ on the interval of non-negative numbers $I$.

For any $n$ and $x$ such that $n<x$, if we choose a $k$ such that $a_{(n-1)+nk}\leq x$ then;

$\Lambda_\chi(x,\alpha_{n,k}) = \frac{n'}{x'} \Phi(x) \pm V$, where the variance $ 0 \leq V \leq \frac{x'-n'}{x'n'} \Phi(x)$

Let $\Lambda(x,n)$ be the the number of totatives of $x$ which are less than or equal to $n$, and $\Phi(x)$ be Euler's totient function.

For now assume $x>n$.

Is there a general formula for $\Lambda(x,n)$? Furthermore, has the result stated below been documented elsewhere?

Let $l = gcd(x,n)$, $x'= x/l$ and $n'=n/l$

I have a wrong (24.09.15) proof that...

$\Lambda(x,n) = \frac{n'}{x'} \Phi(x) \pm V$, where the variance $ 0 \leq V \leq \frac{x'-n'}{x'n'} \Phi(x)$

Extra

Actually from what I can gather this result can be generalized further as follows:

Let $\alpha_{n,k}$ be the interval $ \lbrace a_{0+nk}, a_{1+nk}, ..., a_{(n-1)+nk} \rbrace$ where;

a) $k$ is some natural number such that we can choose its value to position this interval somewhere on the non-negative integer line,

b) $a_i=i$ for all non-negative integers $i$

c) $n$ is a natural number, that we can chose as the size of this interval. (Notice this interval always starts with a number divisible by $n$.)

And, let $\Lambda_\chi(x,I)$ be the number of totatives of $x$ on the interval of non-negative numbers $I$.

For any $n$ and $x$ such that $n<x$, if we choose a $k$ such that $a_{(n-1)+nk}\leq x$ then;

$\Lambda_\chi(x,\alpha_{n,k}) = \frac{n'}{x'} \Phi(x) \pm V$, where the variance $ 0 \leq V \leq \frac{x'-n'}{x'n'} \Phi(x)$

Let $\Lambda(x,n)$ be the the number of totatives of $x$ which are less than or equal to $n$, and $\Phi(x)$ be Euler's totient function.

For now assume $x>n$.

Is there a general formula for $\Lambda(x,n)$  ?

Let $l = gcd(x,n)$, $x'= x/l$ and $n'=n/l$

I have a proof in my head that...

$\Lambda(x,n) = \frac{n'}{x'} \Phi(x) \pm V$, where the variance $ 0 \leq V \leq \frac{x'-n'}{x'n'} \Phi(x)$

Let $\Lambda(x,n)$ be the the number of totatives of $x$ which are less than or equal to $n$, and $\Phi(x)$ be Euler's totient function.

For now assume $x>n$.

Is there a general formula for $\Lambda(x,n)$? Furthermore, has the result stated below been documented elsewhere?

Let $l = gcd(x,n)$, $x'= x/l$ and $n'=n/l$

I have a proof that...

$\Lambda(x,n) = \frac{n'}{x'} \Phi(x) \pm V$, where the variance $ 0 \leq V \leq \frac{x'-n'}{x'n'} \Phi(x)$

Extra

Actually from what I can gather this result can be generalized further as follows:

Let $\alpha_{n,k}$ be the interval $ \lbrace a_{0+nk}, a_{1+nk}, ..., a_{(n-1)+nk} \rbrace$ where;

a) $k$ is some natural number such that we can choose its value to position this interval somewhere on the non-negative integer line,

b) $a_i=i$ for all non-negative integers $i$

c) $n$ is a natural number, that we can chose as the size of this interval. (Notice this interval always starts with a number divisible by $n$.)

And, let $\Lambda_\chi(x,I)$ be the number of totatives of $x$ on the interval of non-negative numbers $I$.

For any $n$ and $x$ such that $n<x$, if we choose a $k$ such that $a_{(n-1)+nk}\leq x$ then;

$\Lambda_\chi(x,\alpha_{n,k}) = \frac{n'}{x'} \Phi(x) \pm V$, where the variance $ 0 \leq V \leq \frac{x'-n'}{x'n'} \Phi(x)$

Let $\Lambda(x,n)$ be the the number of totatives of $x$ which are less than or equal to $n$, and $\Phi(x)$ be Euler's totient function.

For now assume $x>n$.

Is there a general formula for $\Lambda(x,n)$ ?

Let $l = gcd(x,n)$, $x'= x/l$ and $n'=n/l$

I have a proof in my head that...

$\Lambda(x,n) = \frac{n'}{x'} \Phi(x) \pm V$, where the variance $ 0 \leq V \leq \frac{x'-n'}{x'n'} \Phi(x)$

.. but unsure if this result is correct.

Let $\Lambda(x,n)$ be the the number of totatives of $x$ which are less than or equal to $n$, and $\Phi(x)$ be Euler's totient function.

For now assume $x>n$.

Is there a general formula for $\Lambda(x,n)$ ?

Let $l = gcd(x,n)$, $x'= x/l$ and $n'=n/l$

I have a proof in my head that...

$\Lambda(x,n) = \frac{n'}{x'} \Phi(x) \pm V$, where the variance $ 0 \leq V \leq \frac{x'-n'}{x'n'} \Phi(x)$