|
The Riemann theta function is a number-theoretic function which is only really used in the derivation of the functional equation for the Riemann xi function.
The Riemann theta function is defined as:
where is the Riemann omega function.
The domain of the Riemann theta function is .
To give an explicit form for the theta function, note that
and so
Thus we have
Riemann showed that the theta function satisfied a functional equation, which was the key step in the proof of the analytic continuation for the Riemann xi function. This has direct consequences for the Riemann zeta function.
|