Riemann Zeta zeros

TheOogy

in the Riemann Zeta function, is it possible to have two complex zeros off the critical strip that both have the same imaginary part?

solamon

you re asking a question that closes the problem of Riemann hypothesis , i see there s no answer for your question yet .Think about it again or reformulate ,or i m just lost with the question

epkid08

That have a different real part? I would think so. For instance, take s=1/4 and s=3/4 and plug them into the functional equation.
[tex] \zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s) \! [/tex]

TheOogy

i'm looking for two complex zeros that both have the same imaginary part but have diffrent real part, non with real part half. it differs from the riemann hypothesis because i don't care for a single zero off the strip, just pairs. MAYBE some one can prove that, like that dude who proved that 40% of the complex zeros have real part half or the guys who proved all the complex zeros are between 0 and 1. i also don't understand what you mean by zeta(1/4) and zeta(3/4) these are not complex and not zeros.

solamon

i m wondering too why 1/4 and 3/4 ,i m wondering about p/2^n while p<2^n so that we stay in [0 1] with p prime.
i just have noticed the 3/4 , 5/4 when i considered if zeta(z) is close to o then instead of considering zeta(z) i plug in zeta(z) i mean zeta{(zeta(z)} and do some crafting just like heaviside has done on distribution ,some approximation and see the expression not for z but for the complex zeta(z) ...there might be something arround go on try if you have time .

epkid08

I was implying that there probably exists a real number t (due to what I mentioned in my first post) which satisfies:

[tex]\Im[\zeta(1/4 + it)] = \Im [\zeta(3/4 + it)]= 0[/tex]

etc. for other values of re(s)

solamon

have you got some proof,or it s not a statement?did you do some calclations?we can work only on proofs ,and close quickly debates because it s mathematics.

TheOogy

i don't have a proof, which is way i'm asking, i thought maybe someone can prove that there cannot be a pair with the same imaginary part but this turns out more complicated then i thought it would be.

solamon

oh yes i see .you can have an intuition that is true too .you might have a suitable answer .the information i got is that there are solutions with imaginary same absolute value .or with imaginary opposite. check [tex]\bar{z}[/tex] when z is a zero .I would like to make sure of your intuition ,but busy temporaly .later .cheers.

TheOogy

conjugates don't have the same imaginary part.

solamon

yes conjugates don t have have same imaginary except reals .
So try to find out about your idea ,have you some was ?

rscosa

Hi!
fortunately all the complex zeros outside of the known strip are negative and even integers, so of course all of them are (complex but real) zeros of the RZF and have the same imaginary part (equal to zero). But the RZF has no zeros with nonzero imaginary part outside of the strip $0<\re z<1$. By the way those zeros are called trivial zeros.

solamon

What s going on here ?it s an hypothesis to be confirmed or rejected .rscosa ?no statements please about the location of the zeros.we re doing mathematics here .it s just an hypothesis then we re not allowed to consider it true.i m sorry but you have not seen what it s about yet .

CRGreathouse

The RH is that there are no nontrivial zeros off the critical *line*. It has been proven that there are no nontrivial zeros off the critical *strip*.

rscosa

Hi!,
there is NOT an official proof but many people is trying hard.

solamon

Rh is very likely to be true . I have verified myself that if it s true there is not a problem still ,on the limit calculations theory .if it s false there is a big problem.
But who just said it s been proven there are no nontrivial zero off the critical line?do you have the proof reference?i would like to have a look because i have not heard yet about it.problems like this are good not for the results of the proof but they develop logic and minds ,they develop the way of walking close to the reality.

CRGreathouse

No one on this thread has said that there are no nontrivial zeros off the critical line. *I* said that there are no nontrivial zeros off the critical strip -- this is well-known.