Identity of a modification of the primorial function

Question

The primorial function $n\#$ is defined as the product of all primes less than or equal to $n$.

If $p_i$ is the $i$th prime, I define a function $P(n)$ such that $P(n) = p_n\#$. E.g., $P(1) =2, P(2)=6, P(3) = 30$, and so on. NOTE: this is NOT the primorial function. $4\# = 3\# = 6$, while $P(3) = 30\neq P(2)= 6$. This is not to be confused with $p_4\#= 7\# = 2\cdot 3\cdot 5\cdot 7 = 2130$

My questions are:

Does $P(n)$ have a known name?
Is some kind of analytic continuation known for $P(n)$, a la the Gamma function for factorials? My intuition says such an analytic continuation would require the use of the Riemann Zeta function, but I don’t know

EDIT: This proposed duplicate seemed promising, but it at least appears to be an estimation. It’s possible I did my math wrong here, but:

$$ \begin{eqnarray} \log(P(2)) = \log(6) =\sum_{k=1}^2 \log p_n &=& \int_2^3 \log k\; d\pi(k)\\ \log(k)\pi(k)\biggr|_{2}^{3}-\int_{2}^{3}\frac1k \pi(k)dk.\\ =2\log(3)-\log(2)-\lim_{c->3}\int_{2}^{c}\frac1k \pi(k)dk \\ = 2\log(3)-\log(2)-\lim_{c->3}\int_{2}^{c}\frac1kdk =\log(3) \neq \log(6) \end{eqnarray} $$

No, it doesn’t. The Chebyshev function has the same annoyance as the original primorial for me: for composite number C, C# = P#, where P is the largest prime less than C. I’m looking for a “compressed” form of that function, where $P(x+1) = p_i*P(x)$ — Breaking Bioinformatics, Nov 10 '21 at 14:50
Can you clarify the function you have in mind? The values you provide just match the usual primorial. How does yours differ? What is the first value in which they disagree? — lulu, Nov 10 '21 at 14:52
here is the usual primorial function. As you see, it never repeats. — lulu, Nov 10 '21 at 14:53
I accidentally deleted the proposed duplicate. It is this As you can see, they are speaking of the product of the first $n$ primes, just as you are. — lulu, Nov 10 '21 at 14:54
No, that’s the sequence. And they say in the sequence definition that the $n$th number of the sequence is written $p_n#$. If the primorial function was what I was looking for, the $n$th number of the sequence would have been written $n$# — Breaking Bioinformatics, Nov 10 '21 at 14:55
Once again: Please provide an example of where your function differs from this. If I define $F(n)$ to be the $n^{th}$ term of that OEIS sequence, is there a value at which $F(n)$ differs from your $P(n)$? — lulu, Nov 10 '21 at 14:57
$F(n) $ is exactly my function. Unfortunately, $F(n)$ equals $p_i#$, NOT $n#$ — Breaking Bioinformatics, Nov 10 '21 at 15:00
Ok...but the linked duplicate obviously refers to my $F(n)$. Hence to your $P(n)$. Anyway, I'm clearly not seeing what issue you are trying to raise. Don't think my comments are helping, so I'll stop writing. Good luck. — lulu, Nov 10 '21 at 15:01
It's "primorial", not "primordial". I edited your title accordingly. Also, avoid the use of * to denote multiplication. That a common practice in programming, not in Mathematics. — jjagmath, Nov 10 '21 at 15:42
Does this answer your question? Interpolating the primorial $p_{n}#$ — amWhy, Nov 20 '21 at 16:45
I already linked that question in my own and explained why it doesn’t work. — Breaking Bioinformatics, Nov 20 '21 at 18:52

score 1 · Answer 1 · answered Nov 10 '21 at 15:52

We have $$P(n) = \prod_{k=1}^n p_k = \prod_{p\le p_n} p$$ so $$\log P(n) = \sum_{p \le p_n}\log p = \theta(p_n)$$ where $\theta$ is the Chebyshev function.

If you are interested in this function you may use the asymptotic expansions known about $\theta(x)$ and $p_n$.

Identity of a modification of the primorial function

1 Answers1