Approximating ζ(s) between 0 and – 1

The Extended Fibonacci approach will not work with respect to approximations for negative values of s.

However in probing the reason for this further we can uncover the fascinating complementary connections that exits as between the Zeta 1 and Zeta 2 functions.

Once again - as I use the terms - Zeta 1 refers to the traditional Riemann zeta function,
i.e. the infinite series, 1^–s  + 2^–s  + 3^–s  + 4^–s  +……..,   (where s is a complex variable of the form a + it).

The Zeta 2 function refers to the complementary type finite function,

1 + s¹  + s²  + s³  +….. + s^{t – 1}

So the zeros of the Zeta 1 function occur when,

ζ(s₁) =  1^{– s1}   + 2^{– s1}    + 3^{– s1}   + 4^{– s1} +……. = 0.

And the zeros of the Zeta 2 function occur when,

ζ(s₂)  =  1 + s₂¹  + s₂²  + s₂³  +….. + s₂^{t – 1}  = 0 (initially when t is prime).

Now the Extended Fibonacci approach relates directly to approximation of positive real values of  s₁ with respect to the Zeta 1 (i.e. Riemann zeta) function.

However we can envisage the corresponding situation where this Fibonacci approach can be extended to negative values.

Once again positive real values for the extended Fibonacci case relate to the general equation,

x^t  –  x^{t – 1}   –  .... –  1

Negative real values then relate to the general equation,

x^t  =  x^{t – 1}   +  .... +  1

However the RHS of this latter equation now relates directly to the Zeta 2 function!

In fact we have moved from a linear notion of number to a new circular notion!

This can be easily illustrated for the simple case of the zeros of

x^t  + x^{t – 1}   +  .... + 1, where t = 2.

Therefore x²  +  x¹   + 1 = 0 with the two solutions  – .5 + .866i and – .5 – .866i respectively (which lie on the unit circle in the complex plane.

And this highlights a crucial point regarding the relationship of positive values of the Riemann zeta function to negative values i.e. ζ(s) to ζ(1 – s) , which are connected through the Riemann functional equation.

What in fact happens here is that the frame of reference changes.

So positive values of s are identified with a linear notion of number. However negative values apply to a distinctive circular notion. This indeed is the reason why solutions of the Riemann zeta function for negative real values of s appear counter-intuitive.

So when s =  – 1, we have

ζ(s) = 1¹ + 2¹ + 3¹ +....... which from the conventional linear perspective diverges to infinity.

However the value of ζ(– 1)  = –1/12. This seems completely counter-intuitive until we realise that this answer relates to a circular rather than linear notion of number.

At an even deeper level, this entails that the understanding of the Riemann zeta function is inherently dynamic and interactive where analytic keep switching with holistic notions of number (as we relate positive values of s with negative values).

So corresponding to the conventional notion of real numbers as sequentially ordered on the number line (in a quantitative manner) is the corresponding rational notion of linear interpretation (in a qualitative manner).

Likewise corresponding to the paradoxical notion of number as symmetrically arranged as equidistant points on the unit circle (in a quantitative manner) is the corresponding holistic intuitive notion of circular logical interpretation (in a qualitative manner). 

However remarkably this latter type of holistic interpretation - that is indirectly expressed in a circular (i.e. paradoxical) rational manner - is not formally recognised in Conventional Mathematics.

Therefore a true understanding of number - indeed of all mathematical relationships - cannot properly take place until this holistic aspect (that is of equal importance) is coherently integrated with its analytic counterpart.

In psychological terms, this will require that the unconscious aspect be fully integrated (with respect to all mathematical relationships) with the conscious aspect of understanding.    

Now the big breakthrough, in relation to approximating values of  ζ(s) came to me through recognition of the nature of the central value for ζ(– .5).

In this regard, I have been greatly helped by the Riemann Zeta Calculator available online.  

So plugging in the value for s of – .5, ζ(– .5) =  – .207866...

It immediately struck me that this approximated very closely (with negative sign) - perhaps - the most mysterious number in Mathematics i.e. the value of iⁱ  = e ^{– π/2}  = .207879...

And this already strongly suggests the new circular notion of number that is involved.

Indeed the very means of calculating all the "circular" nos. (as various roots of 1) is through the Euler Identity (that is closely related) i.e  e ^{i π}  = cos π + i sin π.

So basically for the estimation of approximations for all other values of s between 0 and 1, s = – .5 serves as the central reference point. 

The formula,
 

– e ^{– π(1/2 + k)} * 2^2k can be conveniently used.

In the default cause where the central value is ζ(– .5), k = 0.

So if we wish to approximate, for example, ζ(– .6), k = .1.

So we obtain – e ^{– π(1/2 + .1)}  * 2^2(.1) 

 

=  – (e ^{– .6π}) *  2^.2  = – .174413...

This compares extremely well with the actual value of  ζ(– .6) = – .174595...

And of course, we are not confined just to values that differ from .5 by .1, .2, .3, .4 or .5.

For example let us calculate now the value for s  =  – .286 i.e. ζ(– .286).

Here, k = – .214

Therefore the approximation ζ(– .286) =  – (e ^{– .286π}) *  2 ^–.428

= .302654...

Once again this compares very favourably with the actual value of ζ(– .286) = .301029.

Like before, I have compiled a table at .1 intervals between 0 and – 1, comparing the approximate values with the actual values of ζ(s) and the % relative accuracy obtained.

s	ζ(s)	Approximation using formula	% accuracy
– .1	– .417228	– .419506	99.5
– .2	– .349666	– .351970	99.3
– .3	– .293813	– .295307	99.5
– .4	– .247165	– .247766	99.8
– .5	– .207866	– .207879	99.99
– .6	– .174595	– .174413	99.9
– .7	– .146237	– .146335	99.9
– .8	– .121987	– .122776	99.4
– .9	– .101193	– .103011	98.2
–1	– .083333	– .086427	96.4

 

Overall, approximations of ζ(s) obtained here of ζ(s) from the exponential formula (involving π) are stunningly accurate. 

The last value represents the famous case of  ζ(–1) = –1/12 which remarkably is approximated by

– 2i²ⁱ.

And  ζ(–1) in the Riemann zeta function represents the sum of the natural numbers! So we are here in the realm of what is most counter-intuitive, from the conventional linear perspective, of number interpretation!