Consider the problem: evaluate the Riemann zeta function . Reminder: for the zeta function is defined

In the rest of the complex plane the zeta function is defined as the analytic continuation (wherever possible). Clearly, we can’t simply put in the above formula. We need to think of something to do the analytic continuation. The starting point is an integral representation…

,

where I inserted unity by means of the Gamma function. We can work further on this

where I substituted . By using the geometric series one gets

.

We can make progress by splitting this integral

,

where

is an entire function. We subtract it from the zeta function and make a Taylor expansion in the denominator:

and then expand the denominator (which is valid in the interval of integration)

Now integrate term by term

Now we’re in business! Note that is an entire function with zeroes at . That means that for the point of interest the term proportional to vanishes and with the help of

we can take the limit

which yields

We also see that the other terms indicated by the ellipses don’t contribute.

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pizza baker, autodidact, particle physicist