If is a multiplicative function, then
Proof of (1).
If the function is defined on prime powers by for all and for all , then allows one to estimate
One of the consequences of this formula is that there are infinitely many primes.
The Riemann zeta function is defined by the means of the series
Since the series converges absolutely, the Euler product for the zeta function is
If we set , then on the one hand is (proof is here (http://planetmath.org/ValueOfTheRiemannZetaFunctionAtS2)), an irrational number, and on the other hand is a product of rational functions of primes. This yields yet another proof of infinitude of primes.
|Date of creation||2013-03-22 14:10:58|
|Last modified on||2013-03-22 14:10:58|
|Last modified by||bbukh (348)|