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December 20[edit]

Approximation of the product of all primes not larger than a given prime.[edit]

Stirling's approximation is an approximation of the product of all integers not larger than a given integer. Is there an approximation of the product of all primes not larger than a given prime? HOTmag (talk) 16:05, 20 December 2021 (UTC)[reply]

What you're looking for is primorial. The characteristics section has a number of approximations.--108.36.85.111 (talk) 16:31, 20 December 2021 (UTC)[reply]

Thanks. So the best approximation discovered so far for the primorial of n is

e^{n}

. HOTmag (talk) 17:02, 20 December 2021 (UTC)[reply]

No, I don't think that's right (either that that's a good approximation, or that it's the best approximation known). If you look at Chebyshev_function you'll see much better (but necessarily more cumbersome) bounds. --JBL (talk) 22:27, 20 December 2021 (UTC)[reply]

That page says

x #

is asymptotically equal to

e (1 + o (1)) x

, so I don't think

e^{x}

is too terrible of an approximation. The other approximations on that page are in terms of k, where

x=p_{k}

, so it's not too clear to me how to get a better closed-form approximation for

x #

from them. Danstronger (talk) 01:54, 21 December 2021 (UTC)[reply]

You'd need very good approximations for the prime-counting function. --Lambiam 08:03, 21 December 2021 (UTC)[reply]

I mean, I guess it depends on what you think constitutes an answer. If you know that the prime you're plugging in is the kth prime, then the bounds on the Chebyshev function give you the actual asymptotic rate of growth (as opposed to exp(n + o(n)), which is only good to within a subexponential multiplicative factor). If you want it as a function of the prime itself, I would have guessed that any error bound on the prime number theorem would translate into a bound on the multiplicative factor (and in particular would let you replace o(n) with something more precise), but I admit to not having worked out the calculation. --JBL (talk) 17:00, 21 December 2021 (UTC)[reply]

Are they talking about this reference^{[20December2021 1]} ?--SilverMatsu (talk) 07:08, 21 December 2021 (UTC)[reply]

References

^ García, E. Muñoz; Marco, R. Pérez (2007). "The Product over All Primes is 4π²". Communications in Mathematical Physics. 277: 69–81. doi:10.1007/s00220-007-0350-z. S2CID 122917501.

[1] García, E. Muñoz; Marco, R. Pérez (2007). "The Product over All Primes is 4π²". Communications in Mathematical Physics. 277: 69–81. doi:10.1007/s00220-007-0350-z. S2CID 122917501.

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