Distribution of a reduced residue system within a primorial

Question

Let $R_{p_k\#}$ be the set of elements in the reduced residue system modulo $p_k\#$.

Let $|R_{p_k\#}|$ be the number of elements in this set.

If $p_i < p_k$ and $p_i$ divides $|R_{p_k\#}|$, does it follow that there are:

$\frac{|R_{p_k\#}|}{p_i}$ elements in $R_{p_k\#}$ inclusive between $1$ and $\frac{p_k\#}{p_i}$

$\frac{|R_{p_k\#}|}{p_i}$ elements in $R_{p_k\#}$ between $\frac{p_k\#}{p_i}$ and $\frac{2p_k\#}{p_i}$

$\dots$

$\frac{|R_{p_k\#}|}{p_i}$ elements in $R_{p_k\#}$ between $\frac{(p_k-1)p_k\#}{p_i}$ and $\frac{(p_k)p_k\#}{p_i}$

For example:

$R_{7\#} = \left\{1, 11, 13, \dots, 209\right\}$ and $|R_{7\#}| = 48$

$3 < 7$ and $3$ divides $|R_{7\#}|=48$ and:

There are $16$ elements between $1$ and $\frac{210}{3} = 70$ which are: $\left\{1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67\right\}$

There are $16$ elements between $70$ and $140$ which are: $\left\{71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139\right\}$

There are $16$ elements between $140$ and $210$ which are: $\left\{143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209\right\}$

Is this always the case? If not, could you provide an example where it is not true? If yes, could you provide the reasoning.

Thanks,

-Larry

Answer 1

I believe that the answer is yes.

Here's my argument:

1. Let $p_i$, $p_k$ be any two primes such that $i < k$

2. Let $R_x$ be the reduced residue system modulo $x$ where $|R_x|$ is the number of elements in $R_x$

3. Assume $p_i$ divides $|R_{p_k\#}|$

4. Let $\varphi(x)$ be Euler's totient function so that we have $\varphi(p_k\#) = |R_{p_k\#}|$

5. $\varphi(\frac{p_k\#}{p_i}) = \frac{\varphi(p_k\#)}{p_i - 1}$

6. So $p_i$ divides $\varphi(\frac{p_k\#}{p_i})$

7. Using the same analysis found in this answer to a previous question, the elements of the reduced residue class modulo $\frac{p_k\#}{p_i}$ divide equally into the congruence classes modulo $p_i$.

8. So, it follows that $\frac{p_i - 1}{p_i}$ of the reduced residue class of $\frac{p_k\#}{p_i}$ are also relatively prime to $p_k\#$.

Answer 2

A counter-example for $p_i$ divides $|R_{p_k\#}|$: the reduced residue system for 2310 has 480 elements in it (calculated using Euler's totient function). It is not true that 7 divides 480 evenly; this would mean that the elements of the reduced residue system modulo primorials are at least sometimes distributed slightly unevenly; however, the Jacobsthal function shows us that the gap between consecutive elements of the reduced residue systems modulo primorials sometimes deviates from the average gap by quite a lot; this sometimes pushes the elements that would have been distributed evenly to the side a bit, creating areas of dense residues and areas of sparse residues, usually right next to each other.

IF they were distributed as evenly as you believed, the twin prime conjecture, De Polignac's conjecture, and the Hardy-Littlewood k-tuple conjecture would easily be proven true and we would have made huge headway towards proving Riemann's hypothesis. As it stands, approaching these ideas through the study of the modular properties of primorials gets extremely close to these proofs, but there seems to be some tiny detail missing from our understanding or which we have not yet connected with these concepts.

EDIT: as stated in the comments, this answer overlooks the fact that the claim was only made for those cases when $p_i$ does divide $\phi(p_k\#)$ (Euler's totient function). I am interested to see what answers others come up with; I personally believe that the study of primorials and their reduced residue systems will bring the most interesting results concerning the distribution of the primes. I will be digging for a while on this, as I hope anyone who reads this will do as well.