Riemann zeta function
Infinite series

The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.

In fact the even and odd terms give the two sums

and

Parametrized versions of the above sums are given by

and

with | t | < 2 {\displaystyle |t|<2} and where ψ {\displaystyle \psi } and γ {\displaystyle \gamma } are the polygamma function and Euler's constant, respectively, as well as

all of which are continuous at t = 1 {\displaystyle t=1}. Other sums include

where Im denotes the imaginary part of a complex number.

There are yet more formulas in the article Harmonic number.
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