Here is a striking example of the unfortunate (but probably unavoidable) dispersion of mathematics: at the end of the abstract of V. Arnold’s talk at the recent conference in honor of A. Douady, one can read:
The Cesaro mean values K̂ of the numbers K(n) tend, as n tends to ∞, to a finite limit K̂(∞)=lim 1/n ∑m=1n K(m) = 15/π2. This theorem, deduced from the empirical observation of the coincidence of 20 first digits, is now proved, using the formula K̂(∞)=ζ(2)/ζ(4)
Here, K(n) is defined before in the abstract as the expression
$latex K(n)=\prod_{p\mid n}{(1+1/p)}$
Arnold’s achievements as mathematician are about as impressive as it can get. But the statement here is a completely elementary exercise in analytic number theory, and has been for at least one century (i.e., Dirichlet, or Chebychev, could do it in a few minutes, if not Euler). Here’s the proof in Chebychev style:
$latex K(n)=\sum_{d\mid n}{\mu(d)^2/d}$
hence, exchaning the sum over n and the sum over d, we get
$latex \sum_{n<X}{K(n)}=\sum_{d<X}{\mu(d)^2d^{-1} [X/d]}$
and replacing the integral part by X/d+O(1), this is clearly asymptotic to CX with
$latex C=\sum_{d\geq 1}{\mu(d)^2d^{-2}}$
which is an absolutely convergent series. As an Euler product it is
$latex \prod_{p}{(1+p^{-2})}=\prod_{p}{\frac{1-p^{-4}}{1-p^{-2}}}=\frac{\zeta(2)}{\zeta(4)}$
as desired.
