i am requesting to unique scholars who are familiar with riemann hypothesis to explain at what point or at which equation we have to prove real part of zeta function is 1/2.
Hi, one way to answer your question is to say that it has been proved that there are infinitely many non-trivial zeros exactly on this critical line. If I remember correctly, it was Hardy who proved this first. So, to verify the hypothesis, it suffices to prove that there are no non-trivial zeros elsewhere.
@ Timo Tossavainen I have two questions : 1° What do you mean by "non-trivial" zeroes ? 2° In the absence of a precise definition, I don't understand the logic of your assertion : "So (...) it suffices to prove that there are no non-trivial zeroes elsewhere".
@ s h s hussainsha Coming back by chance to your question, I feel that some clarification is needed. You ask "at what point or what equation we have to prove real part of Zeta function is 1/2 in Riemann hypothesis" ? The "equation" is Zeta(s) = 0 (in analytic number theory, the complex variable is denoted s), so your "points" are but the zeroes of Riemann's Zeta function, and the RH states that all the "non trivial" zeroes should have real part equal to 1/2. It remains to define whtat a "non trivial" zero is.
Recall that the series with general term n-s converges absolutely for Re(s) >1, and Zeta (s) is its sum in this domain. To get Zeta(s) in the whole complex plane, you have to show (this is a theorem) that the function just defined can be extended to a meromorphic function on C, with only one pole at s=1, which is a simple pole with residue 1. This is this meromorphic continuation which is called Riemann's Zeta.
To go further, you need to show a "functional equation" for Zeta. This is an explicit identity relating the values of Zeta at s and 1-s. I cannot type it here, but you can find it in any textbook.This functional equation shows immediately that for any integer m > 0, Zeta(s) has a simple zero at s= -2m. These zeroes are called the "trivial zeroes" of Zeta. So finally, RH states that the non trivial" zeroes of Zeta are all located on the line Re(s) = 1/2 . Note that this "critical line" is a symmetry axis for the functional equation.
NB: Your sentence "real part of Zeta function is 1/2" is not correct.
- Badges
- Science topic
- Similar topics
- Mathematics