More Interesting Relationships

Here are some interesting relationships, which I discovered some time ago in relation to the Riemann Zeta Function (for positive integers > 1).

 ∑{ζ(s) – 1} ~ 1 (for s = 2, 3, 4,…..)

For example from adding up values for s = 2 to 10, we obtain

.6449 + .20205 + .08232 + .03692 + .0173 + .00834 + .00407 + .002008 + .000904

= .99812 (which is already close to 1)

Then  ∑{ζ(s) – 1} ~ .75 (for even values of s i.e. s = 2, 4, 6, …)

So for even values of s from 2 to 10, we obtain

.6449 + .08232 + .0173 + .00407 + .000904

= .749494 (which again is very close to .75).

Also ∑{ζ(s) – 1} ~ .25 (for odd values of s i.e. 3, 5, 7, ….)  

So for odd values of s from 3 to 10, we obtain

.20205 + .03692 + .00834 + .002008

= .249318 (which for just 4 values computed is again close to .25)  

There are also interesting connections as between the Riemann zeta function (for positive integer values of s and the Euler- Mascheroni constant i.e. γ = .5772156649…

As is well known for ζ(s) where s = 1 (i.e. the harmonic series) and the summation of the series is taken over a finite set of values n,

ζ(1) = log n + γ

However γ in turn is related to all ζ(s) - now summed without limit - for the other positive integer values of s in the following manner!

γ = ζ(2)/2 – ζ(3)/3 + ζ(4)/4 – ζ(5)/5 + ……

So for s = 2 to 10, we obtain

1.644934/2 – 1.202056/3 + 1.082323/4 – 1.03692/5 + 1.0173/6 – 1.00834/7 + 1/00407/8 – 1.002008/9 + 1.000904/10

= .62474….

Now this approximation is still not very accurate, but in this case the series the series diverges very slowly towards the true value (oscillating alternating above and below the true value).

A better approximation however can be obtained as follows:

1 – γ  = {ζ(2)/2 – 1}/2 + { ζ(3)/3 – 1}/3 + {ζ(4)/4 – 1}/4 + {ζ(5)/5 – 1}/5 + ……

So again summing for s = 2 to 10, we obtain

.644934/2 + .202056/3 + 082323/4 + .03692/5 + .0173/6 + .00834/7 + .00407/8 + .002008/9 + .000904/10

= .42268 (correct to 5 decimal places) which gives γ = .57732 which is already a very good approximation to the true value i.e. .5772156649…

Also ζ(s)/ζ(s + 1)   ~ 1, and

{ζ(s) – 1}/{ζ(s + 1) – 1}   ~ 2, again for sufficiently large t.

For example ζ(9) = 1.002008 and ζ(10) = 1.000904

Therefore ζ(9)/ ζ(10) = 1.002008/1.000904 = 1.0011… (which is already close to 1)

Likewise ζ(9) – 1 = .002008 and ζ(10) – 1 = .000904

Therefore {ζ(9) – 1}/{ζ(10) – 1} = .002008/.000904 = 2.2212…

This is not yet very close to 2. However for larger t the ratio will progressively fall towards 2!

In all cases i.e. for positive integers > 1, ζ(s) can be expressed as 1 + k (where k is less than 1)

So for example ζ(2) = 1.6449… = 1 + .6449…

We can then define a “complementary” number as 1 – k

So in the case of  ζ(2), 1 – k = 1 – .6449… = .3551

We can now define a new set of twin relationship as π^s/t_s1 = 1 + k and π^s/t_s2 = 1 – k respectively.

t_s1 and t_s2 new are the two denominators associated with the common numerator π^s.

For example when s = 2,  π²/6 = 1 + .6449… and π²/27.79… = 1 – .6449… respectively.

So here, t_s1 = 6 and t_s2 = 27.79… respectively

And the difference of t_s2  and t_s1  = 27.79 – 6 = 21.79…

When s grows sufficiently large {t_s2  –  t_s1}/{t_{(s + 1)2}  –  t_{(s + 1)1}}  ~ 2/π

For example when s = 9, k = .002008; t_s1 = 29749 (to nearest integer) and t_s2 = 29869.

Therefore t_s2  – t_s1   = 120.

When s = 10, k = .000904; t_{(s + 1)1}  = 93555 and t_{(s + 1)2}  = 93733 

Therefore t_{(s + 1)2}  –  t_{(s + 1)1}   = 178

So {t_s2  –  t_s1}/{t_{(s + 1)2}  –  t_{(s + 1)1}}  = 120/178 = .6741…

This compares fairly well with 2/π = .6366..

And the approximation steadily improves for larger s.

Approximating the Non-Trivial Zeros (2)

Having approximated the first 10 of the non-trivial zeros, I decided to continue on an calculate the first 30.

Once again I am used the slightly modified formula i.e. t/2π(log t/2π –  1) + 1.

And as there are 29 non-trivial zeros up to 100, this means that we have thereby approximated all the non-trivial zeros for t to 100!

However in the original approximation of values, where I adjusted the first calculation for each zero downward (by half the deviation from the next value), a bias still remained in that the overall sum of the actual zeros tended to be consistently overshoot that of the corresponding approximations. Therefore in the attempt to eliminate this bias I decided to use a new adjustment factor (based on the deviations of the 1st set of approximations).

Therefore to more accurately approximate the nth zero, I decided to multiply the deviation as between the nth and (n + 1)st value by (1 – 2/π) and then subtract this from the original 1st approximation.

So below, I have provided a table showing the three different approximations, together with the acual values for the trivial zeros.

I have then highlighted the most recent approximations and actual values in bold type for easier comparison.

Riemann Zeros	Predicted Location (1)	Deviation of Zeros	Predicted Location (2)	Predicted Location (3)	Actual Location
1st	17.08		14.34	15.09	14.13
2nd	22.56	5.48	20.27	20.90	21.02
3rd	27.14	4.58	25.09	25.65	25.01
4th	31.24	4.10	29.34	29.86	30.43
5th	35.04	3.80	33.28	33.76	32.94
6th	38.56	3.52	36.88	37.34	37.59
7th	41.92	3.36	40.28	40.73	40.92
8th	45.20	3.28	43.64	44.06	43.33
9th	48.33	3.13	46.82	47.23	48.01
10th	51.36	3.03	49.89	50.29	49.77
11th	54.31	2.95	52.87	53.26	52.97
12th	57.19	2.88	55.78	56.17	56.45
13th	60.00	2.81	58.62	59.18	59.35
14th	62.76	2.76	61.40	61.78	60.83
15th	65.47	2.71	64.14	64.51	65.11
16th	68.12	2.65	66.81	67.17	67.08
17th	70.74	2.62	69.45	69.80	69.55
18th	73.32	2.58	72.05	72.40	72.07
19th	75.86	2.54	74.61	74.95	75.70
20th	78.36	2.50	77.12	77.46	77.14
21st	80.83	2.47	79.60	79.94	79.34
22nd	83.28	2.45	82.07	82.40	82.91
23rd	85.70	2.42	84.50	84.83	84.74
24th	88.09	2.40	86.91	87.23	87.43
25th	90.46	2.37	89.29	89.61	88.81
26th	92.80	2.34	91.64	91.95	92.49
27th	95.12	2.32	93.96	94.28	94.65
28th	97.43	2.31	96.28	96.60	95.87
29th	99.72	2.29	98.59	98.90	98.83
30th	101.98	2.26	100.86	101.17	101.32
31st	104.22	2.24

Once again, I consider it striking how the simple general formula provides such a convenient means for calculating, with stunning accuracy, not only the frequency of zeros up to any given t, but likewise a ready means for approximating the value for each one of the trivial zeros.

The difference as between the actual values for the zeros and their corresponding approximations is due to the local random nature of the zeros.

However this randomness is at the other extreme from the primes. In fact both the primes and trivial zeros complement each other in a dynamic interactive manner.

So the behaviour of individual primes is as independent as possible consistent with maintaining an overall collective interdependence with each other (through the natural numbers).

However the collective behaviour of the non-trivial zeros is as interdependent (i.e. ordered) as possible, consistent with each zero maintaining an individual local independence.

Therefore whereas the simple general formula for frequency of  primes can only hope to predict with a strictly relative degree of accuracy, the corresponding formula for frequency of non-trivial zeros can predict in absolute terms with a remarkable level of accuracy.

Approximating the Non-Trivial Zeros

So far we have concentrated on approximations of ζ(s) for real values of s.

Of course, the Riemann zeta function is also defined for complex values of s (a + it). However apart from the hugely important complex values of s for which ζ(s) = 0, i.e. the famed Riemann zeros, I am not going to suggest further approximation formulas.

It is postulated according to the Riemann Hypothesis that all complex solutions for ζ(s) = 0, lie on the line through .5 and are of the form .5 + it and .5 – it respectively.

So we will concentrate here on the values of t for which ζ(s) = 0.

I had already approached this issue in an earlier blog entry "Using Frequency Formula to Estimate Location of Riemann Zeros"

The approximation here was based on  realisation of the amazing accuracy of the formula that is used to calculate the frequency of zeros up to a given t.

This formula is given as t/2π(log t/2π –  1). However, I found that a very slight adjustment (through addition of 1) gave an even better estimate.

So the formula I am using is  t/2π(log t/2π –  1) + 1.

In fact in all the examples I calculated up to numbers in the region of 10²², this formula either yielded exactly the correct frequency (in absolute terms) or was out by at most 1.

This then suggested to me a way for attempting to approximate the value of each zero by successively inserting the integer values 1, 2, 3, 4,.... representing frequency of zeros as the result form the formula, and then working backwards to find the value of t to which these results related.

I then provided the estimates I thereby obtained - correct to 2 decimal places - for the first 10 values for t in this manner.

However following from the initial observation that the first t thus calculated (17.08) was very close to exactly half-way as between the 1st and 2nd actual zeros, I then realised that a much closer approximation could be obtained by adjusting these values (using the deviations existing between the approximation results). 

So for example with respect to the 1st estimated zero, the idea is to subtract half of the deviation as between 1st and 2nd calculated values to better approximate the result for this 1st zero.

Therefore we obtain 17.08 - 2.74 = 14.34 which indeed compares very favourably with the actual result (i.e. 14.13).

I have now drawn up a new table which provides these latest approximations (with % relative accuracy) for the first 10 zeros.

Riemann Zeros	Predicted Location (1)	Deviation of zeros	Predicted Location (2)	Actual Location	% accuracy
1st	17.08		14.34	14.13	98.5
2nd	22.56	5.48	20.27	21.02	96.4
3rd	27.14	4.58	25.09	25.01	99.7
4th	31.24	4.10	29.34	30.43	96.4
5th	35.04	3.80	33.28	32.94	99.0
6th	38.56	3.52	36.88	37.59	98.1
7th	41.92	3.36	40.29	40.92	98.5
8th	45.18	3.26	43.60	43.33	99.4
9th	48.34	3.16	46.84	48.01	97.6
10th	51.34	3.00	49.96	49.77	99.6
11th	54.31	2.97

As can be seen, the predicted (approximate) results are accurate to a surprising level of accuracy.

The nature of the Riemann zeros complements in a dynamic interactive manner that of the prime nos.

Whereas the behaviour of the primes is locally as independent as possible (compatible with an overall collective relationship with the natural numbers being maintained), it is somewhat the reverse with the Riemann zeros. The very nature of these zeros is to smooth out as much as possible local incompatibilities as between the quantitative (independent) and qualitative (interdependent) nature of the primes, thus entailing a continuous regularity enabling the precise calculation of their frequency (in absolute terms) to an extraordinary degree of accuracy.