## 10. The ζ (zeta) function

In the previous sections we have seen how, given a real function of real variable, one can find its series expansion about 0; it is inversely possible define a real function giving an infinite convergent sum. This is the way L. Euler used to define the ζ function for real variables. After him, B. Riemann extended the domain of this function to complex numbers. Now this function is known as the Riemann's ζ function. Here, however, we consider only real arguments.

The Riemann's ζ (zeta) function is defined as If x=1, the (10.1) is the harmonic series, which don't converge. So for x=1 the function has a singularity.

The values of ζ(n), when n is an even natural number, can be exactly deduced. The method, found by L. Euler, may be the following.

• Starting from the MacLaurin series expansion of the sine when x≠0, one can get p(x) has the same zeroes of sin x, excluding 0.

• In general, a polynomial P(x), with non-null zeroes xk and such that P(0)=1, may be factorized as Therefore p(x) may be written • To fast the calculation, let ; we get The MacLaurin series expansion of p(α) is • The first derivative of p(α) may be calculated using the logarithmic derivative method For α=0 we get p'(0)α must coincide with , therefore Finally we get the value of ζ(2) • From p'(α) we can have p''(α)  must coincide with , so From that we get the value of ζ(4) • From p''(α) we can get p'''(α)  must coincide with , therefore Finally, the value of ζ(6) is These values of ζ are strictly correlated with the Bernoulli's numbers; they are particular cases of a general relation true for all the natural even numbers: From this relation we have also The extension of the domain of ζ to the real and complex number shows that also the integer negative arguments generate values related to the Bernoulli's numbers. It can be demonstrated that, for natural numbers n≥2, that is The Bernoulli's numbers with odd index ≥ 3 are all 0, so the values of ζ for negative even integer values are all 0.

For arguments other than 1, the values of ζ can be numerically approximated (P. Borwein in "An Efficient Algorithm for the Riemann Zeta Function").

The following JavaScript application allows you to approximate the values of Z for real arguments. If your browser does not allow the `iframe` tag, you can open the source page.