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July 9

Riemann zeta function

From Riemann zeta function#Euler product formula we have:

The connection between the zeta function and prime numbers was discovered by Euler, who proved the identity

\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}},

where, by definition, the left hand side is

ζ (s)

and the infinite product on the right hand side extends over all prime numbers

p

....

Both sides of the Euler product formula converge for

Re(s) > 1

.... Since the harmonic series, obtained when

s = 1

, diverges, Euler's formula ... implies that there are infinitely many primes.

For convergent cases, I understand this. But for the case s=1, what does this mean other than the vacuous ∞ = ∞ ? Does it mean that if you take both sides for the first k terms and take their ratio, that this ratio converges to 1 as k goes to infinity, or that their difference converges to 0, or what? Loraof (talk) 03:10, 9 July 2017 (UTC)[reply]

For s=1 both sides are infinite, but it's not exactly vacuous since it proves that there are an infinite number of primes. The idea of the ratio converging to 1 or difference converging to zero doesn't really make sense since the sum and product are taken over different sets. You are right in that infinity isn't technically a number, so saying two infinite limits are equal is a bit of a shortcut and manipulating diverging a series/product requires a bit more care to be strictly rigorous, something Euler would not have worried about. You could make it a bit more formal as follows: For a given M there is k so that

\sum _{n=1}^{k}{\frac {1}{n}}>M

. Using the argument given to prove the s>1 case, it's easy to see that

\prod _{p{\text{ prime and }}p\leq k}{\frac {1}{1-p^{-1}}}>M

since when you multiply out the product it contains every term in the sum. So the limit of this as k→∞ is infinity. This implies the number of factors goes to infinity as k→∞, in other words the number of primes is infinite. Note that something stronger is true, namely that

\sum _{p{\text{ prime}}}{\frac {1}{p}}

diverges; this is the real improvement of Euler over Euclid.--RDBury (talk) 07:05, 9 July 2017 (UTC)[reply]

Thanks, RDBury. Sorry if I'm just being slow here, but what do you mean by "when you multiply out the product it contains every term in the sum"? Loraof (talk) 16:49, 9 July 2017 (UTC)[reply]

For example,

{\begin{aligned}&{\frac {1}{1-2^{-1}}}\,{\frac {1}{1-3^{-1}}}\,{\frac {1}{1-5^{-1}}}\\&=\left(1+{\frac {1}{2}}+{\frac {1}{4}}+\dots \right)\left(1+{\frac {1}{3}}+{\frac {1}{9}}+\dots \right)\left(1+{\frac {1}{5}}+{\frac {1}{25}}+\dots \right)\\&=1+{\frac {1}{2}}+{\frac {1}{3}}+\dots \end{aligned}}

where the sum on the right runs over natural numbers whose prime factors are restricted to 2. 3 and 5. This includes all terms from 1/1 to 1/5 as well as some others. --RDBury (talk) 18:21, 9 July 2017 (UTC)[reply]

@RDBury: Note that something stronger is true, namely that $\sum _{p{\text{ prime}}}{\frac {1}{p}}$ diverges; this is the real improvement of Euler over Euclid. Huh, if true that is certainly stronger, but I fail to see how it follows from what you wrote. Tigraan^{Click here to contact me} 17:01, 12 July 2017 (UTC)[reply]
- Our article divergence of the sum of the reciprocals of the primes (which I wish was at a title like prime harmonic series) gives the (admittedly questionable) way Euler derived the result from there. Double sharp (talk) 07:49, 13 July 2017 (UTC)[reply]