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As a math amateur, I am finding the study reduced residue systems relative a primorial a very interesting way to understand the distribution of primes. For example, it is fascinating to me that it is so easy to count the number of integers $x < p\#$ where $\gcd(x(x+2),p\#)=1.$

When I do a google search on reduced residue systems, I find very little that is interesting. On this site, for example, I see only 12 results.

What is the reason that there seems to be little active research on reduced residue systems modulo a primorial? Is it that the material is so well-studied that it no longer has much appeal? Is it that other areas seem so much more promising? Is it that research in reduced residue systems relative primorial is now focused on highly technical topics?

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The Chinese remainder theorem tells us that the residue class ring ${\bf Z}/p\# {\bf Z}$ is isomorphic (as a ring) to the product of the finite fields ${\bf Z}/q {\bf Z}$, where $q$ ranges over the primes up to $p$. As such, many "global" or "multiplicative" questions about the residue classes modulo a primorial quickly reduce to questions about residue classes modulo a prime $q$. The latter type of question is quite interesting and is indeed intensively studied (see e.g. the immense literature surrounding the Weil conjectures). However, the passage from the prime (or prime powers) case to products of primes tends to be substantially less interesting, and often simply amounts to multiplying together the answers obtained from each individual prime.

If one wants to return to number theory on the integers themselves, rather than the residue class ring ${\bf Z}/p\# {\bf Z}$, then one typically needs to understand the "local" behaviour of reduced residue systems and similar sets, in which one only looks at what is going on in a relatively small subinterval of this ring (e.g. the residue classes between $p$ and $p^2$). For instance, the recent progress on finding large gaps between primes (see e.g. these two papers) relies on finding somewhat large gaps inside the reduced residue system of a primorial. However, the global product structure of ${\bf Z}/p\# {\bf Z}$ is often not all that effective for studying these local problems, and techniques coming from sieve theory, combinatorics, or analysis tend to be more useful.

As to why your literature searches return so few hits, I think it is in part because terminology used by researchers in number theory has diverged over time from that used by the amateur number theory community, perhaps due to the influence of other modern areas of research mathematics such as algebra or geometry. For instance, the term "primorial" is not often used in research literature (one may write for instance $\prod_{p \leq w} p$ or $P(w)$ in place of $w\#$). Residue class rings ${\bf Z}/q{\bf Z}$ are often referred to by other names, such as cyclic groups or reductions of the integers modulo $q$, and the reduced residue system might be referred to as the group of units or primitive residue classes.

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  • $\begingroup$ Thanks very much! This is exactly the answer that I was looking for. $\endgroup$ – Larry Freeman Apr 26 '15 at 5:59

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