Number theoretical considerations lead to the modification of zeta function by replacing the powers n-s with= exp(-log(n)s) with powers exp(-Log(n)s), where rational valued number theoretic logarithm Log(n) is defined as sumpp p/π(p) corresponding to the decomposion of n to a product of powers of prime. For large primes Log(p) equals in good approximation to log(p). The point of the replacement is that Log(n) carriers number theoretical information so that the definition is very natural. This number theoretical zeta will denoted with Ζ to distinguish it from ordinary zeta function denoted by ζ.
It is interesting to list the elementary properties of the Ζ before trying to see whether functional equation for ζ and Riemann hypothesis generalize.
The key questions are whether Ζ allows a generalization of the functional equation ξ(s)= ξ(1-s)
with ξ(s)= (1/2) s(s-1) Γ(s/2) π-s/2 ζ(s) and whether Riemann hypothesis generalizes. The derivation of the functional equation is quite a tricky task and involves integral representation of ζ .
- The replacement log(n)→ Log(n)== sump kpLog(p) implies that Ζ codes explicitly number theoretic information. Note that Log(n) satisfies the crucial identity Log(mn)= Log(m)+ Log(n). Ζ is an analog of partition function with rational number valued Log(n) taking the role of energy and 1/s that of a complex temperature. In ZEO this partition function like entity could be associated with zero energy state as a "square root" of thermodynamical partition function: in this case complex temperatures are possible.|Ζ|2 would be the analog of ordinary partition function.
- Reduction of Ζ to a product of "prime factors" 1/[1-exp(-Log(p)s)] holds true by Log(n)== sump kpLog(p), Log(p) =p/π(p).
- Ζ is a combination of exponentials exp(-Log(n)s), which converge for Re(s)>0. For ζ one has
exponentials exp(-log(n)s), which also converge for Re(s)>0: the sum ∑ n-s does not however converge in the region Re(s)<1. Presumably Ζ fails to converge for Re(s)≤ 1. The behavior of terms exp(-Log(n)s) for large values of n is very similar to that in ζ.
- One can express ζ o in terms of η function defined as
η(s)= ∑ (-1)n n-s .
The powers (-1)n guarantee that η converges (albeit not absolutely) inside the critical strip 0<s<1.
By using a decomposition of integers to odd and even ones, one can express ζ in terms of η:
ζ = η(s)/(-1+2-s+1) .
This definition converges inside critical strip. Note the pole at s=1 coming from the factor.
One can define also Η as counterpart of η:
Η(s)= ∑ (-1)n e-Log(n)s) .
The formula relating Ζ and Η generalizes: 2-s is replaced with exp(-2s) (Log(2)=2):
Ζ = Η(s)/(-1+2e-2s) .
This definition Ζ converges in the critical strip Re(s) ∈ (0,1) and also for Re(s)>1.
Ζ(1-s) converges for Re(s)<1 so that in Η representation both converge.
Note however that the poles of ζ at s=1 has shifted to that at s=log(2)/2 and is below Re(s)=1/2 line. If a symmetrically posioned pole at s= 1-log(2)/2 is not present in Η, functional equation cannot be true.
- Log(n) approaches log(n) for integers n not containing small prime factors p for which π(n) differs strongly from p/log(p). This suggests that allowing only terms exp(-Log(n)s) in the sum defining Ζ not divisible by primes p<pmax might give a cutoff Ζcut,pmax behaving very much like ζ from which "prime factors" 1/(1-exp(-Log(p)s) , p<pmax are dropped of. This is just division of Ζ by these factors and at least formally, this does not affect the zeros of Ζ. Arbitrary number of factors can be droped. Could this mean that Ζcut has same or very nearly same zeros as ζ at critical line? This sounds paradoxical and might reflect my sloppy thinking: maybe the lack of the absolute implies that the conclusion is incorrect.
There is however no need for a functional equation if one is only interested in F(s)== Ζ(s)+Ζ(1-s) at the critical line! Also the analog of Riemann hypothesis follows naturally!
- One can start from the integral representation of ζ true for s>0.
ζ(s)=[1/(1-21-s)Γ(s)]∫0∞[ts-1/(et+1)] dt , Re(s)>0 .
deducible from the expression in terms of η(s). The factor 1/(1+et) can be expanded in geometric series 1/(1+et)=∑ (-1)n exp(nt) converning inside the critical strip. One formally performs the integrations by taking nt as an integration variable. The integral gives the result ∑ (-1)n/nz)Γ(s).
The generalization of this would be obtained by a generalization of geometric series:
1/(1+et)=∑ (-1)n exp(nt)→ ∑ (-1)n eexp(Log(n))t
in the integral representation. This would formally give Ζ: the only difference is that one takes u= exp(Log(n))t as integration variable.
One could try to prove the functional equation by using this representation. One proof (see this) starts from the alternative expression of ζ as
ζ(s)=[1/Γ(s)]∫1∞[ ts-1/(et-1)]dt , Re(s)>1 .
One modifies the integration contour to a contour C coming from +∞ above positive real axis, circling the origin and returning back to +∞ below the real axes to get a modified representation of ζ:
ζ(s)=1/[2isin(π s)Γ(s)]∫1∞[(-w)s-1/(ew-1)] dw , Re(s)>1 .
One modifies C further so that the origin is circle d around a square with vertices at +/- (2n+1)π and +/- i(2n+1)π.
One calculates the integral the integral along C as a residue integral. The poles of the integrand proportional to 1/(1-et) are at imaginary axis and correspond to w= ir2π, r∈ Z. The residue integral gives the other side of the functional equation.
- Could one generalize this representation to the recent case? One must generalize the geometric series defined by 1/(ew-1) to -∑ eexp(Log(n))w. The problem is that one has only a generalization of the geometric series and not closed form for the counterpart of 1/(exp(w)-1) so that one does not know what the poles are. The naive guess is that one could compute the residue integrals term by term in the sum over n. An equally naive guess would be that for the poles the factors in the sum are equal to unity as they would be for Riemann zeta. This would give for the poles of n:th term the guess wn,r=r2π/exp(Log(n), r∈ Z. This does not however allow to deduce the residue at poles.Note that the pole of Η at s= log(2)/2 suggests that functional equation is not true.
To sum up, the definition of modified zeta and eta functions makes sense, as also the analog of Riemann Hypothesis. It however seems that the counterpart of functional equation does not hold true. This is however not a problem since one can define symmetrized zeta so that it is well-defined in critical strip.
- In the representation using Η F(s) converges at critical striple and is real(!) at the critical line Re(s)=1/2 as follows from the fact that 1-s= s* for Re(s)=1/2! Hence F(s) is expected to have a large number of zeros at critical line. Presumably their number is infinite, since F(s)cut,pmax approaches 2ζcut,pmax for large enough pmax at critical line.
- One can define a different kind of cutoff of Ζ for given nmax: n<nmax in the sum over e-Log(n)s. Call this cutoff Ζcut,nmax. This cutoff must be distinguished from the cutoff Ζcut,pmax obtained by dropping the "prime factors" with p<pmax. The terms in the cutoff are of the form u∑ kpp/π(p), u = exp(-s). It is analogous to a polymomial but with fractional powers of u. It can be made a polynomial by a change of variable u→ v=exp(-s/a), where a is the product of all π(p):s associated with all the primes involved with the integers n<nmax.
One could solve numerically the zeros of Ζ(s)+Ζ(1-s) using program modules calculating π(p) for a given p and roots of a complex polynomial in given order. One can check whether also all zeros of Ζ(s)+Ζ(1-s) might reside at critical line.
- One an define also F(s)cut,nmax to be distinguished from F(s)cut,pmax. It reduces to a sum of terms exp(-Log(n)/2) cos(-Log(n)y) at critical line, n<nmax. Cosines come from roots of unity. F(s) function is not sum of rational powers of exp(-iy) unlike Ζ(s). The existence of zero could be shown by showing that the sign of this function varies as function of y. The functions cos(-Log(n)y) have period Δ y = 2π/Log(n). For small values of n the exponential terms exp(-Log(n)/2) are largest so that they dominate. For them the periods Δ y are smallest so that one expected that the sign of both F(s) and F(s)cut,nmax varies and forces the presence of zeros.
One could perhaps interpret the system as quantum critical system. The rather large rapidly varying oscillatory terms with n<nmax with small Log(n) give a periodic infinite set of approximate roots and the exponentially smaller slowly varying higher terms induce small perturbations of this periodic structure. The slowly varying terms with large Log(n) become however large near the Im(s)=0 so that here the there effect is large and destroys the period structure badly for small root of Ζ.
For background see the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title.