Primorial

In mathematics, and more particularly in number theory, primorial, denoted by " p_{n}\# ", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors.

In mathematics, and more particularly in number theory, primorial, denoted by " p_{n}\# ", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors. p_n\# = \prod_{k=1}^n p_k, where p_k is the th prime number. For instance, p_5\# signifies the product of the first 5 primes: p_5\# = 2 \times 3 \times 5 \times 7 \times 11= 2310. The first few primorials p_n\# are: 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690... . Asymptotically, primorials grow according to p_n\# = e^{(1 + o(1)) n \log n}. n\# = \prod_{p\,\leq\, n\atop p\,\text{prime}} p = \prod_{i=1}^{\pi(n)} p_i = p_{\pi(n)}\#, where \pi(n) is the prime-counting function . This is equivalent to n\# = \begin{cases} 1 & \text{if }n = 0,\ 1 \\ (n-1)\# \times n & \text{if } n \text{ is prime} \\ (n-1)\# & \text{if } n \text{ is composite}. \end{cases} For example, 12\# represents the product of all primes no greater than : 12\# = 2 \times 3 \times 5 \times 7 \times 11= 2310. Since \pi(12)=5 , this can be calculated as: 12\# = p_{\pi(12)}\# = p_5\# = 2310. Consider the first 12 values of the sequence : 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310. We see that for composite , every term n\# is equal to the preceding term . In the above example we have 12\# = p_5\# = 11\# since is composite. Primorials are related to the first Chebyshev function \vartheta(n) by \ln (n\#) = \vartheta(n). Since \vartheta(n) asymptotically approaches n for large values of , primorials therefore grow according to: n\# = e^{(1+o(1))n}. For any , n\#=p\# iff p is the largest prime such that . Let p_k be the th prime. Then p_k\# has exactly 2^k divisors. The sum of the reciprocal values of the primorial converges towards a constant \sum_{p\,\text{prime}} {1 \over p\#} = {1 \over 2} + {1 \over 6} + {1 \over 30} + \ldots = 0{.}7052301717918\ldots The Engel expansion of this number results in the sequence of the prime numbers. Griffiths (2015) proved that it is irrational. Euclid's proof of his theorem on the infinitude of primes can be paraphrased by saying that, for any prime p , the number p\# +1 has a prime divisor not contained in the set of primes less than or equal to p . . For , the values are smaller than e , but for larger n , the values of the function exceed e and oscillate infinitely around e later on. Since the binomial coefficient \tbinom{2n}{n} is divisible by every prime between n+1 and , and since \tbinom{2n}{n} \leq 4^{n} , we have the following upper bound: n\#\leq 4^n . Using elementary methods, Denis Hanson showed that . Using more advanced methods, Rosser and Schoenfeld showed that n\#\leq (2.763)^n . Furthermore, they showed that for , . \zeta(k)=\frac{2^k}{2^k-1}+\sum_{r=2}^\infty\frac{(p_{r-1}\#)^k}{J_k(p_r\#)},\quad k\in\Z_{>1} .

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