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.