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You can read the question above. So I'm really " new in terms of Riemann hypothesis". I have read about the hypothesis in wikipedia. So I know the statement of the Hypothesis : The Riemann Zeta Function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. but I dont understand why the statement in the title is an accord for this hypothesis? $q$ is the number of elements in $F$, a finite field.

Thank you for explanation.

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    $\begingroup$ "I know that the Riemann Zeta Function has its zeros only at the negative even integers and complex numbers with real part $\frac12$." - Actually, nobody else knows that $\endgroup$ – Hagen von Eitzen Jan 9 at 19:47
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    $\begingroup$ This follows just by a change of variables. This gives the direct analogue of RH (which is for the usual zeta function), i.e., the fact that the roots of $Z_f(u)$ have absolute value $q^{-(n-1)/2}$ is equivalent to the fact that the roots of $\zeta_f(s)$ have real part $1/2$. $\endgroup$ – Dietrich Burde Jan 9 at 19:50
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    $\begingroup$ Very poor choice of words, Hagen von Eitzen. I meant I know the statement of the Hypothesis. @DietrichBurde oh. Interesting. Do you have a source where I can see the change of variables for your claim ? :) $\endgroup$ – RukiaKuchiki Jan 9 at 20:02
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    $\begingroup$ The change is $u=q^{-s}$. See "Weil conjectures" in Wikipedia, or at Terry Tao's blog, among other references. You should perhaps clarify your question and give a link to the article you were reading. $\endgroup$ – Dietrich Burde Jan 9 at 20:08
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    $\begingroup$ You should look at the local and global zeta and L-functions of elliptic curves (over finite fields and over $\mathbb{Q}$) and search about the analytic continuation, functional equations and Riemann hypothesis of each one (the local and global Riemann hypothesis are analytically unrelated, by you'll see why the wording are the same and how certain concepts transfer from the local to the global case) $\endgroup$ – reuns Jan 9 at 20:17

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