How does one distinguish between the trivial and non-trivial zeros on the critical line in the Riemann Hypothe

Question

On the complex plane, there are both trivial and non-trivial zeros, or solutions to the Zeta function. I am not sure how to distinguish between them. Anone have a quick one or two sentence explanation?

Answer 1

The trivial zeros are at (-2K+0i), where K is a positive integer. The nontrivial zeros are all the others. It has been shown that the nontrivial zeroes must have a real part between 0 and 1.

Answer 2

The Riemann zeta function has zeros on the adverse even integers (see the clever equation). those are referred to as the trivial zeros. they are trivial interior the experience that their life is comparatively effortless to instruct, as an occasion, from sin(?s/2) being 0 interior the clever equation. The non-trivial zeros have captured lots extra interest with the aid of fact their distribution no longer in user-friendly terms is a lot much less understood yet, extra importantly, opens a thank you to an astonishingly wealthy vein of mathematical inquiry. it truly is known that any non-trivial 0 lies interior the open strip {s ? C: 0 < Re(s) < a million}, this is named the serious strip. The Riemann hypothesis, seen to be between the main suitable unsolved problems in arithmetic, asserts that any non-trivial 0 s has Re(s) = a million/2. interior the theory of the Riemann zeta function, the set {s ? C: Re(s) = a million/2} is named the serious line. basically, I haven't any friggin' theory... good day JEFF S! Congrats on the Orange Badge!! (your über-deleted chum Evil Elton)

Answer 3

The trivial zeros are -2n for n integral > 0.
The non-trivial zeros are the other complex
zeros of the zeta function and are conjectured to all lie
on the critical line x == 1/2.

Answer 4

Pascal is right about the trivial zeroes but the hypothesis about the non trivial zeroes having a real part between 0 and 1 has not been proven and a proof of it (The Riemann hypothesis) is worth one million dollars. See www.claymath.org under Millenium Awards.

Answer 5

Look at the critical time-line function as defined in the zeta-rumien tandien theorem. The bypass matrix denotes a well defined triviality test.