I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\s...

The zeta function is defined for an argument with real part greater than unity by the series. ζ (z) = ∑ k = 1 ∞ 1 k z = 1 + 1 2 z + 1 3 z + ⋯ Historically the first connection between the zeta function and trigonometry occurred in the process of solving the Basel problem, which was to evaluate the infinite sum of inverse squared integers

Mar 8, 2021 · ζ is the Riemann Zeta function. From Mittag-Leffler Expansion for Cotangent Function: Factoring − 1 k2: Taking |z| <1, and noting that k ≥ 1, we have, by Sum of Infinite Geometric Sequence: from which:

Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$

Nov 29, 2014 · It is not hard to show, using Liouville's theorem, that. π cot(πz) = 1 z + ∑n=1∞ (1 z + n + 1 z − n), which implies that. −πz 2 cot(πz) = −1 2 +∑k=1∞ ζ(2k)z2k, 0 <|z| <1. This formula predicts correctly that ζ(0) = −1 2, and allows to calculate ζ(2k) as a rational multiple of π2k as well (in terms of Bernoulli numbers).

We certainly have $|\pi \cot(\pi z)|<2\pi$ for $z$ on the circle and $R$ large enough. The term $1/z^{2n}$ is $\Theta(1/R^{2n})$ so that the integral along the circle is $\Theta(1/R^{2n-1})$ and vanishes in the limit.

The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane.

That is, z’(z) is a generating function for the Euler{Riemann zeta function (k) at positive even values of k. On the other hand, the second expansion is essentially a generating function for the Bernoulli numbers. Again for jzj<1, ˇzcotˇz= ˇiz+ 2ˇiz e2ˇiz 1 = ˇiz+ X1 k=0 B k k! (2ˇiz)k = 1 + X even k 2 (2ˇi)kB k k! zk:

1. The Riemann zeta function The Riemann zeta function ζ : (1,∞) → (0,∞) is defined by the formula ζ(s) = X∞ k=1 k−s. The sum on the right-hand side is by definition the limit limn Pn k=1k −s of the partial sums; the limit exists (though it may be ∞) because the partial sums form a non-decreasing sequence of (positive) real ...

Expression of cot via (2 ) cot = 1 − 2 ∑︁∞ =1 2 2 2 1 − 2 2 . From the geometric series formula, 2 2 1 − 2 2 = ∑︁∞ =1 2 2 2 , so there is cot = 1 − 2 ∑︁∞ =1 ∑︁∞ =1 2 2 2 = 1 −2 ∑︁∞ =1 2 −1 2 ∞ =1 1 2 | {z } (2 ).