formulae for zeta in the critical strip
Let us use the traditional notation for the complex variable, where and are real numbers.
where denotes the largest integer , and denotes .
Lemma: For integers and such that :
Proof: If we can prove the special case , namely
so the right side of (4) is
and the lemma is proved.
Now take and let in the lemma, showing that (2) holds for . By the principle of analytic continuation, if the integral in (2) is analytic for , then (2) holds for . But is bounded, so the integral converges uniformly on for any , and the claim (2) follows.
is analytic on . Write
and integrate by parts:
The first two terms on the right are zero, and the integral converges for because is bounded.
Remarks: We will prove (1) in a later version of this entry.
Using formula (3), one can verify Riemann’s functional equation in the strip . By analytic continuation, it follows that the functional equation holds everywhere. One way to prove it in the strip is to decompose the sawtooth function into a Fourier series, and do a termwise integration. But the proof gets rather technical, because that series does not converge uniformly.
|Title||formulae for zeta in the critical strip|
|Date of creation||2013-03-22 13:28:14|
|Last modified on||2013-03-22 13:28:14|
|Last modified by||mathcam (2727)|