Editing Riemann–Siegel theta function (section)

==Gram points==
The Riemann zeta function on the critical line can be written

:<math>\zeta\left(\frac{1}{2}+it\right) = e^{-i \theta(t)}Z(t),</math>
:<math>Z(t) = e^{i \theta(t)} \zeta\left(\frac{1}{2}+it\right).</math>

If <math>t</math> is a [[real number]], then the [[Z function]] <math>Z(t)</math> returns ''real'' values.

Hence the zeta function on the critical line will be ''real'' either at a zero, corresponding to <math>Z(t)=0</math>, or when 
<math>\sin\left(\,\theta(t)\,\right)=0</math>. Positive real values of '''<math>t</math>''' where the latter case occurs are called '''Gram points''', after [[Jørgen Pedersen Gram|J. P. Gram]], and can of course also be described as the points where <math>\frac{\theta(t)}{\pi}</math> is an integer.

A '''Gram point''' is a solution <math>g_n</math> of
:<math>\theta(g_n) = n\pi.</math>

These solutions are approximated by the sequence:

:<math>g'_n = \frac{2 \pi \left(n + 1 - \frac{7}{8}\right)}{W\left(\frac{1}{e} \left(n + 1 - \frac{7}{8}\right) \right)},</math>

where <math>W</math> is the [[Lambert W function]].

Here are the smallest non negative '''Gram points'''
{| class="wikitable" border="1"
|-
! <math>n</math>
! <math>g_{n}</math>
! <math>\theta(g_{n})</math>
|-
|style="text-align:right;"| −3
|style="text-align:right;"|  0
|style="text-align:right;"| 0
|-
|style="text-align:right;"| −2
|style="text-align:right;"|  3.4362182261...
|style="text-align:right;"| −{{pi}}
|-
|style="text-align:right;"| −1
|style="text-align:right;"|  9.6669080561...
|style="text-align:right;"| −{{pi}}
|-
|style="text-align:right;"| 0
| 17.8455995405...
|style="text-align:right;"| 0
|-
|style="text-align:right;"| 1
| 23.1702827012...
|style="text-align:right;"| {{pi}}
|-
|style="text-align:right;"| 2
| 27.6701822178...
|style="text-align:right;"| 2{{pi}}
|-
|style="text-align:right;"| 3
| 31.7179799547...
|style="text-align:right;"| 3{{pi}}
|-
|style="text-align:right;"| 4
| 35.4671842971...
|style="text-align:right;"| 4{{pi}}
|-
|style="text-align:right;"| 5
| 38.9992099640...
|style="text-align:right;"| 5{{pi}}
|-
|style="text-align:right;"| 6
| 42.3635503920...
|style="text-align:right;"| 6{{pi}}
|-
|style="text-align:right;"| 7
| 45.5930289815...
|style="text-align:right;"| 7{{pi}}
|-
|style="text-align:right;"| 8
| 48.7107766217...
|style="text-align:right;"| 8{{pi}}
|-
|style="text-align:right;"| 9
| 51.7338428133...
|style="text-align:right;"| 9{{pi}}
|-
|style="text-align:right;"| 10
| 54.6752374468...
|style="text-align:right;"| 10{{pi}}
|-
|style="text-align:right;"| 11
| 57.5451651795...
|style="text-align:right;"| 11{{pi}}
|-
|style="text-align:right;"| 12
| 60.3518119691...
|style="text-align:right;"| 12{{pi}}
|-
|style="text-align:right;"| 13
| 63.1018679824...
|style="text-align:right;"| 13{{pi}}
|-
|style="text-align:right;"| 14
| 65.8008876380...
|style="text-align:right;"| 14{{pi}}
|-
|style="text-align:right;"| 15
| 68.4535449175...
|style="text-align:right;"| 15{{pi}}
|}

The choice of the index ''n'' is a bit crude. It is historically chosen in such a way that the index is 0 at the first value which is larger than the smallest positive zero (at imaginary part 14.13472515 ...) of the Riemann zeta function on the critical line. Notice, this <math>\theta</math>-function oscillates for absolute-small real arguments and therefore is not uniquely invertible in the interval [−24,24]. Thus the [[Odd function|odd]] theta-function has its symmetric Gram point with value 0 at index −3.
Gram points are useful when computing the zeros of <math>Z\left(t\right)</math>. At a Gram point <math>g_n,</math>

:<math>\zeta\left(\frac{1}{2}+ig_n\right) = \cos(\theta(g_n))Z(g_n) = (-1)^n Z(g_n),</math>

and if this is ''positive'' at ''two'' successive Gram points, <math>Z\left(t\right)</math> must have a zero in the interval.

According to '''Gram’s law''', the [[real part]] is ''usually'' positive while the [[imaginary part]] alternates with the Gram points, between ''positive'' and ''negative'' values at somewhat regular intervals.

:<math>(-1)^n Z(g_n) > 0</math>

The number of roots, <math>N(T)</math>, in the strip from 0 to ''T'', can be found by
:<math>N(T) = \frac{\theta(T)}{\pi} + 1+S(T),</math>
where <math>S(T)</math> is an error term which grows asymptotically like <math>\log T</math>.

Only if <math>g_n</math> '''would obey Gram’s law''', then finding the number of roots in the strip simply becomes

:<math>N(g_n) = n + 1.</math>

Today we know, that in the long run, '''Gram's law''' fails for about 1/4 of all Gram-intervals to contain exactly 1 zero of the Riemann zeta-function. Gram was afraid that it may fail for larger indices (the first miss is at index 126 before the 127th zero) and thus claimed this only for not too high indices. Later Hutchinson coined the phrase ''Gram's law'' for the (false) statement that all zeroes on the critical line would be separated by Gram points.