Zeta Constant - Negative Integers

Negative Integers

In general, for negative integers, one has

for .

The so-called "trivial zeros" occur at the negative even integers:

The first few values for negative odd integers are

However, just like the Bernoulli numbers, these do not stay small for increasingly negative odd values. For details on the first value, see 1 + 2 + 3 + 4 + · · ·.

So can be used as the definition of all (including those for index 0 and 1) Bernoulli numbers.