Gram Points

In numerical analysis of the Riemann zeta function ζ(s) one uses θ defined as:

where the argument is defined by continuous variation of t starting with the value 0 at t = 0. For large t, θ has the asymptotic expansion

For n ≥ 1, the n−th Gram point g_n is defined as the unique solution > 7 to θ(g_n) = nπ. The Gram points are as dense as the zeros of ζ(s) but are much more regularly distributed. Their locations can be found without any evaluations of the Riemann-Siegal series. In locating the Riemann zeta zeros one studies the the rotated function Z(t) = exp(iθ(t))γ(1/2 + it).

In analogy with Gram points, we can define generalized Gram points on the critical axis as follows:

Definition 2.1. For t ≥ 7, t is said to be a generalized Gram point with value φ if θ(t) = 2kπ + φ, where 0 ≤ φ < 2π.

Universality of Riemann Zeta value distribution on critical axis

An interesting paper presents a remarkable and striking universality property for the Riemann zeta value distribution at generalized Gram points. The work shows excellent agreement of the actual value distribution with the universality relations prediction. More detail.

The properties generalize for Dirichlet L Functions, see Value Distribution of Dirichlet L function on critical axis.

Gram's Law

Gram's law is the empirical observation that Z(t) usually changes its sign in each Gram interval Gn = [g_n , g_n+1 ). This law fails infinitely often, but it is true in a large proportion of cases. The average value of Z(g_n) is 2 for even n and −2 for odd n, and hence Z(g_n) undergoes an infinite number of sign changes.

Conjectures regarding Riemann zeta function at Gram points

See https://www.tandfonline.com/eprint/sspr9k4ZZNbWEwMj2qZj/full?target=10.1080/10586458.2018.1492474

On this page we study the properties of the distribution of the rotated zeta function (Hardy’s function) Z(g_n) at gram points. We present a couple of conjectures. These conjectures are most likely related to the symmetry properties of the value distribution of the Riemann

zeta function. The conjectures are as follows.

Conjecture 1 (even-odd antisymmetry): The distribution of the zeta values for odd gram points is the negative of the distribution of zeta values for the even zeta points.

Conjecture 2 (forward-backward symmetry): When we consider a sequence of zeta values at consecutive gram points, the properties of the sequence are symmetric with respect to the direction of the sequence of gram points (i.e., the sequence behaves similarly whether we consider the points in increasing order or in decreasing order).

For further discussion see "Conjectures".

For conjecture 1, compare the figures below for the distribution of zeta values for the even gram points and the odd gram points.

Distribution of zeta for 50000 even gram points starting at gram point 1,000,000,000,000

Normal Q-Q plot for 50000 even gram points starting at gram point 1,000,000,000,000

Distribution of zeta for 50000 odd gram points starting at gram point 999,999,999,999 Normal Q-Q plot for 50000 odd gram points starting at gram point 999,999,999,999

Sequence of zeta values for 100000 gram points starting at gram point 999,999,999,999.

The distribution of zeta values considering both odd and even gram points is given below. The distribution is not normal, there is a sharp peak and a long tail of extreme values at both ends.

Distribution of zeta for 100000 gram points starting at gram point 999,999,999,999 Normal Q-Q plot for 100000 gram points starting at gram point 999,999,999,999

Further validation of the conjectures comes from the statistical parameters of the distributions (quantiles, mean, median).

> summary(r$zeta) (all gram points)

Min. 1st Qu. Median Mean 3rd Qu. Max.

-129.400 -1.166 0.00084 0.01206 1.17500 109.20

> summary(e$zeta) (even gram points)

Min. 1st Qu. Median Mean 3rd Qu. Max.

-67.330 0.1263 0.8623 2.0120 2.5460 109.20

> summary(o$zeta) (odd gram points)

Min. 1st Qu. Median Mean 3rd Qu. Max.

-129.400 -2.5310 -0.8618 -1.9880 -0.1248 57.36

Other properties of Gram Points

Good and Bad Gram Points

We will follow the treatment and notation of Xavier Gourdon.

A Gram point g_n is called good if (−1)ⁿZ(g_n) > 0, and bad otherwise. A Gram block is an interval [g_n, g_n+k) such that g_n and g_n+k are good Gram points and g_n+1, . . ., g_n+k−1

are bad Gram points. A Gram block is denoted by the notation a₁a₂ . . . a_k where k is called the length of the Gram block, and a_i denote the number of roots of Z(t) in the Gram interval

[g_n+i−1, gn+i). So far, no Gram interval has been found with more than 5 zeros, thus the notation is unambiguous. a₁ and a_k must be even while a₂ to a_k-1 are odd.

Regular Gram Block

A Gram block of length k which contains exactly k roots of Z[t} is called regular. The first and last Gram intervals of a regular Gram block must contain an even number of roots (0 or 2 roots). The internal Gram intervals must all contain an odd number of zeros (all of them must contain one zero if the end intervals contain 2 and 0 roots. If the end intervals both contain no zeros, then one of the internal intervals must contain 3 zeros.)

Thus, in the notation of Gourdon, Regular Gram blocks must have a pattern of one of the following three forms:

21 . . . 10, 01 . . . 12, 01 . . . 131 . . . 10.

where the notation 1 . . . 1 refers to any string of consecutive 1, including zero length string. Gourdon denotes these three types of regular Gram Blocks as Type I, Type II and Type III respectively. A simple generalization of Conjecture 2 above predicts that the number of Type I and Type II Gram blocks up to a given large height must be equal to each other (except for Type I Gram blocks of length one, which only have the pattern 1). The table below (from Gourdon) shows the number of regular Gram Blocks of different types found upto t = 10¹³. The prediction of Conjecture 2 is borne out to a remarkable degree. Further validation comes from a consideration of the different types of violations of Rosser's rule, which also show the forward-backward symmetry.

Critical Behaviour for Riemann zeta function? Sharp Transition

While there is very good evidence for the forward-backward symmetry from the ratio of Type II to Type I counts, there is another intriguing pattern: the Type II counts are ever so slightly smaller than the Type I counts (See table below). Thus, while we have a symmetry, it is broken very slightly. This would remind a particle physicist of the weak interactions!

Rosser's rule

Rosser's rule states that Gram blocks of length k contain at least k zeros. This law is violated infinitely often, but is violated only for a small fraction of the Gram Blocks. A Gram block is either regular, or it has an excess of zeros, or it has fewer zeros than its length. The last type of Gram block violates Rosser's rule.

If a Gram block of length k is an exception to Rosser rule, then its pattern of zeros must be of the form 01...10. To describe the exception, we must specify where the two missing zeros are. Gourdon uses the notation

kXa1 ...am, X = L or R

to describe an exception on a Gram block of length k where the missing zeros are on the left (for X = L) or on the right (for X = R), the pattern containing the missing zeros being a1 . . . am (moreover, this pattern is the smallest union of Gram block adjacent to the exception that contains the missing zeros). For example, 3L04 denotes a violation of Rosser rule on a Gram block of length 3, the missing zeros being at its left. Globally, the pattern of zeros expressed by the notation is 04010.

The notation above classifies the type of violation of Rosser rule, the value m being called the length of the excess block. The notation used for exceptions to Rosser rule is not unambiguous. When several contiguous violations of Rosser rule exists, they may overlap or missing zeros can be in the same Gram interval. Such situations are very rare, and in these cases (Gourdon found just three occurrences until the 1013-th zero), Gourdon uses the notation Ma1 ...al where the pattern a1 . . . al is made of the minimal contiguous Gram blocks containing at least one violation to Rosser rule, and all the missing zeros. For example, the pattern M00500, first encountered at gram index n = 3, 680, 295, 786, 518, denotes a situation with two violations of Rosser rule (“00” and “00”, Gram blocks with missing zeros) and a single Gram interval containing all the missing zeros (pattern “5”).

See Also Riemann zeta zero counts on Gram Intervals