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.  

Approximation Formulas for Negative Values of ζ(s)

We have concentrated on the separate tasks of approximating the value of ζ(s) for s between 0 and
– 1 and – 1 and – 2 respectively.

However it is not very fruitful to attempt to continue in estimating ζ(s) for further negative values of s in this somewhat piecemeal manner.

Some years ago, I formulated a number of approximation formulae for Negative Odd Integers of the Riemann Zeta Function for making such calculations. These are basically of the form that based on one known value an other can then be approximated.

One such formula is given as:

ζ (s – 2)/ζ(s) →  .1 + 11k/2π² + k(k – 3)/π²

Now for this calculation, we use absolute values of ζ(s - 2) and ζ(s) and k in the formula is given as k = – (s + 1)/2.

So when s = – 1,  k = 0.

Therefore  ζ – 3)/ζ(– 1) →  .1

So, given that ζ(– 1) = 1/12 (in absolute terms), therefore  ζ – 3) →  1/120.

However the perfect accuracy in this case represents beginner's luck, as approximations generally will not be exact.

For example if we were to continue to attempt to calculate ζ – 5) from  ζ – 3),

k =  – (– 3 + 1)/2 = 1.

Therefore  ζ – 5)/ζ(– 3)  →  .1 + 11/2π² –2/π² = .45462.

So the absolute value of  ζ – 5) → .45462(1/120) → 1/264.

Though not exact, this approximates well the true value 1/252 and the relative approximation steadily improves for larger values of s. 

In fact, a more accurate approximation can come from using our original approximation for ζ(– 1) i.e. – .086427.

This would then give the approximation for ζ – 3) as  – .0086427 and the corresponding approximation for ζ – 5) as  .003929... = 1/254.5 in absolute terms which is 99% accurate.


And we can also use this formula for non-integer values.

For example let us attempt to calculate the value of ζ – 2.5) from our already obtained approximation for  ζ – 0.5)  i.e. – .207879.

k – (s + 1)/2 = – .25

Therefore ζ – 2.5)/ζ(– 0.5)  →  .1 –2.75/2π² + .8125/π² = .04300
   

So in absolute terms ζ – 2.5) → .043 * 207879 = .008173.

This compares well with the correct answer .008516... (96% relative accuracy).

So once we have the means of approximating all values for  ζs) in the range for s from 0 to – 2, then we can in principle ultimately approximate ζs) for all negative values of s using this formula.


However, there are other methods of approximation, that I suggested at that time.


Once again where s is a negative odd number, this alternative formula can be used,

{│ζ(s - 2k)}^{1/(1 – s + 2k)}│}/{│ζ(s)}^{1/(1 – s)}│}  →  (1 - s + 2k)/(1 – s)

When we confine ourselves to negative integer values of s, the value for k = 1.

Therefore in this special case, 

{│ζ(s - 2)}^{1/(3 – s)}│}/{│ζ(s)}^{1/(1 – s)}│}  →  (3 - s)/(1 – s)

This is especially useful for larger absolute values of s.

For example, given knowledge of │ζ – 13)│  = 1/12, let us consider the approximation that can be obtained for ζ – 15).

So {│ζ(- 15)}^1/16│} /{│ζ(- 13)}^1/14│} →  16/14

Thus  {│ζ(- 15)}^1/16│} →  16/14 * {│ζ(- 13)}^1/14│}

→ 16/14 * .837366 = .956990

Therefore │ζ(-15)│ →.4949

This  compares with the true value = 3617/8160 = .4434...

This still is not very exact. However the relative accuracy of the approximation continually improves for larger absolute values of s