Gram Points

The Gram points $g_n$ form an infinite sequence of real numbers where real part the Riemann zeta function $\zeta(\frac{1}{2}+s)$ function vanishes as it crosses the critical line. A real number number $g_n$ is a Gram point if:

$$\vartheta(g_n) = g_n\pi$$

where $\vartheta(t)$ is the argument of the Riemann-Siegel theta function which links the Riemann zeta function to the Hardy Z function via the Riemann-Siegel formula.

$$\zeta\left(\frac{1}{2}+it\right) = e^{-i \theta(t)}Z(t)$$

$$Z(t) = e^{i \theta(t)} \zeta\left(\frac{1}{2}+it\right)$$

If $t$ is a real number, then the Z function $Z(t)$ returns real values.

Hence the zeta function on the critical line will be real-valued when

$$\sin ( \theta(t) ) = 0$$

Positive real values of $t$ where this occurs are called Gram points, after Jørgen Pedersen Gram, and can of course also be described as the points where $$\frac{\theta(t)}{\pi}$$ is an integer.

A Gram point is a solution $g_n$ of $$\theta(g_n) = n\pi$$

These solutions are approximated by the sequence:

$$g'_n = \frac{2 \pi \left(n + 1 - \frac{7}{8}\right)}{W\left(\frac{1}{e} \left(n + 1 - \frac{7}{8}\right) \right)},$$

where $W$ is the Lambert W function.

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 $\theta$-function oscillates for absolute-small real arguments and therefore is not uniquely invertible in the interval [−24,24]! Thus the odd theta-function has its symmetric Gram point with value 0 at index −3. Gram points are useful when computing the zeros of $Z(t)$. At a Gram point it is true that

$$\zeta\left(\frac{1}{2}+ig_n\right) = \cos(\theta(g_n))Z(g_n) = (-1)^n Z(g_n)$$

and if this is positive at two successive Gram points, $Z(t)$ must have a zero in the interval.

Gram's one quarter observation

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

$(-1)^n Z(g_n) > 0$

The number of roots, $N(T)$, in the strip from 0 to $%$, can be found by

$$N(T) = \frac{\theta(T)}{\pi} + 1+S(T)$$

where $S(T)$ is an error term which grows asymptotically like $\log T$

Only if $g_n$ would obey Gram’s rule, then finding the number of roots in the strip simply becomes

$$N(g_n) = n + 1$$

Today we know, that in the long run, Gram's law is not true for about 1/4 of all Gram-intervals to contain exactly 1 zero of the Riemann zeta-function. Gram thought that it may not be true for larger indices (the first skip 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. However, it is still an important concept when studying the zeros of the Riemann zeta function and the Hardy Z-function.