The Reimann Zeta Function

An enlightening discussion about pi and tau.

The Reimann Zeta Function

by 1=0 » Wed Apr 26, 2017 7:30 pm

Currently, the most important unsolved problem in mathematics is the Reimann zeta hypothesis. Doing a little research on this problem, you will find many, many formulas simplified by tau, and barely any simplified by pi. This provides a much better argument for tau than an archaic problem which only uses one formula does for pi.

The reason the zeta hypothesis is important in the first place is because of the relationship between prime numbers and the non-trivial zeroes of the zeta function. This relationship is expressed in the “explicit formula” as
Σ(x^ρ/ρ)+ln(τ)+1/2ln(1-x^(-2))+ψ(x)=x
ρ
where ρ is a non-trivial zero, x is greater than one and not a prime power, and ψ is the Mangoldt function (a function related to primes).
This is not the only identity involving the Reimann zeta function that is simpler with tau. For a positive even integer n, ζ(n)=|Bn|τ^n/(2n!)
And the derivative at -n is ζ’(-n)=(-1)^(n/2)ζ(n+1)n!/(2τ^n)
When S±(n) is the sum from k=1 to infinity of 1/[k^(-n)(e^(τk)±1)], odd values of ζ(n) can be expressed in terms of τ^n and S±(n) with rational coefficients. There are two formulas that find the odd values of n, both of which are simplified by tau in multiple places. However, they are too complicated for me to type into this comment, so I'll just link to them
http://mathworld.wolfram.com/images/equ ... tion42.gif
http://mathworld.wolfram.com/images/equ ... tion43.gif
The positive integer values of the function can also be found with a integral expressions which are simpler with tau (which I will also just link to)
http://mathworld.wolfram.com/images/equ ... tion37.gif
http://mathworld.wolfram.com/images/equ ... ine283.gif
The latter is for odd integers and also has three slight variations which are also simpler with tau.
Some of the derivatives of the Reimann zeta function are
ζ’(0)=-1/2lnτ
ζ’’(0)=γ[1]+1/2γ^2-τ^2/96-½(lnτ)^2
ζ’’’(0)=3lnτγ[1]+3γγ[1]+3/2γ[2]-ζ(3)-½(lnτ)^3-τ^2/32lnτ+3/2γ^2lnτ+γ^3
where γ is the Euler-Mascheroni constant but γ[n] are Stieltjites constants.
ζ’(-2n)=(-1)^nζ(2n+1)(2n)!/(2τ^n)
ζ’(2)=ζ(2)(γ+lnτ-12lnA) and ζ’(-1)=1/12-lnA, where A is the Glaisher-Kinkelin constant. Multiple definitions of A are simpler with tau.
The sum from 1 to infinity of ζ(2n)/[n(2n+1)] is lnτ-1 and a similar formula involving the Hurwitz zeta function is http://mathworld.wolfram.com/images/equ ... tion49.gif
The functional equation for the zeta function is ζ(1-s)=2τ^(-s)cos(sτ/4)Γ(s)ζ(s).
The Reimann zeta function can be defined as a contour integral: http://mathworld.wolfram.com/images/equ ... tion16.gif
The Z-function is related to the critical line of the Reimann zeta function and can be expressed as https://wikimedia.org/api/rest_v1/media ... 76d17a574d
where R(t) is asymptotically expressed in terms of https://wikimedia.org/api/rest_v1/media ... 725b2dd104 and u=(t/τ)^0.25
Possibly the most important fact about the Z-function is its density of real zeroes, which is also the density of non-trivial zeroes of the Reimann zeta function if the Reimann hypothesis is true. The density is c/τln(t/τ)
There is a generalization of the Reimann zeta function called the Hurwitz zeta function. A few formulas involving the Hurwitz zeta function are
http://mathworld.wolfram.com/images/equ ... ation6.gif
http://mathworld.wolfram.com/images/equ ... ation7.gif
http://mathworld.wolfram.com/images/equ ... ation8.gif
http://mathworld.wolfram.com/images/equ ... tion14.gif
The can be even further generalized with the Lerch zeta function and the Lerch transcendent (which also have multiple other important functions as special cases). The Lerch zeta function is https://wikimedia.org/api/rest_v1/media ... 8cb1c39fd7
The Lerch transcendentcan be calculated in multiple ways:
https://wikimedia.org/api/rest_v1/media ... 725c7c0f7c
https://wikimedia.org/api/rest_v1/media ... 81cfd0fbd7
https://wikimedia.org/api/rest_v1/media ... 2c2e09c158
And its asymptotic series include
https://wikimedia.org/api/rest_v1/media ... 25bf1ae8e6
https://wikimedia.org/api/rest_v1/media ... db69756754
The Reimann zeta function can also be generalized as the periodic zeta function: http://mathworld.wolfram.com/PeriodicZetaFunction.html

With an insanely long list of examples of how tau simplifies the most important conjecture currently eluding mathematicians and next to no formulas involving the zeta function that pi simplifies, how can pi compete with tau in terms of simplicity?
τ>π
1=0
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