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Given the Riemann-Siegel function, why are the Gram points important? I say if we have $S(T)$, the oscillating part of the zeros, then given a Gram point and the imaginary part of the zeros (under the Riemann Hypothesis), are the Gram points near the imaginary part of the Riemann zeros?

I say that if the difference $ |\gamma _{n}- g_{n} | $ is regulated by the imaginary part of the Riemann zeros.

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If you don't start accepting answers, people just won't help you. – sxd Nov 12 '11 at 14:05
but in comments there are not thick mark for votes :) .. whenever they gave a question i click on the thick mark if possible i will review all the answers given and will vote again. – Jose Garcia Nov 13 '11 at 11:33
This is not about comments. You can upvote comments, but you can't accept them (and people don't get reputation for upvotes to their comments). It's about accepting one of the answers to your question. Answers and comments are two separate things. Answers are the things underneath where it says "$n$ Answers:". Comments are the things in small print underneath the question and the answers. – joriki Nov 13 '11 at 15:01
up vote 7 down vote accepted

One thing Gram points are good for is that they help in bracketing/locating the nontrivial zeroes of the Riemann $\zeta$ function.

More precisely, recall the Riemann-Siegel decomposition


where $Z(t)$ and $\vartheta(t)$ are Riemann-Siegel functions.

$Z(t)$ is an important function for the task of finding nontrivial zeroes of the Riemann $\zeta$ function, in the course of of verifying the hypothesis. (See also this answer.)

Riemann-Siegel function

That is to say, if some $t_k$ satisfies $Z(t_k)=0$, then $\zeta\left(\frac12+it_k\right)=0$.

Now, Gram points $\xi_k$ are numbers that satisfy the relation $\vartheta(\xi_k)=k\pi$ for some integer $k$. They come up in the context of "Gram's law", which states that $(-1)^k Z(\xi_k)$ tends to be positive. More crudely, we can say that Gram points tend to bracket the roots of the Riemann-Siegel function $Z(t)$ (i.e. there is often a root of $Z(t)$ in between consecutive $\xi_k$):

Gram points

"Gram's law" doesn't always hold, however:

A bad Gram point

There are a number of other "bad" Gram points.

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