## Friday, October 21, 2022

ξ function (see this) is closely related to ζ and is much simpler. In particular, it lacks the trivial zeros forcing to drop from ζ the Euler factor to get ζp. ξ has a very simple representation completely analogous to that for polynomials (see this):

ξ(s)=(1/2)∏k (1-s/sk )(1-s/(sk*) .

Only the non-trivial zeros appear in the product.

1. For s=O(p), this product is finite but need not converge to a well-defined p-adic number in the infinite extension of p-adic numbers. Also the values of ξ (s) at integer points are known to be transcendental so that the interpretation as a generalization of a rational polynomial fails. Note that the presence of an infinite number terms in the product can cause transcendentality of the coefficients of ξ(s). Algebraic numbers are required. ξ(2n) is proportional to π2 and ξ(2n+1) to ζ(2n+1)/π2. The presence of an infinite number of terms in the expansion of ξ(s) can however cause this.
2. If ξ is indeed analogous to a rational polynomial, the analog of discriminant

D=∏k≠ l(sk-sl)(sk-sl*)

should be proportional to a product of powers of ramified primes for ξ. The simplest option would be that there is complete number theoretic democracy and the ramification is minimal. If so, D would be proportional to the product Y= ∏k pk of all primes, appearing in the definition of infinite primes (see this). This number is infinite in real sense but would have p-adic norms 1/pk.

3. The Hadamard product can be written in the form

ξ(s)=(1/2)∏k (1+ s(s-1)/Xk) , Xk= sks*k ,

in which the s↔ 1-s symmetry is manifest. The power series of ξ(s)== ξ1(u)=∑ an un, u=s(s-1), should converge for all primes p.

If regard s(s-1) as p-adic number and apply the inverse of I s(s-1) to get real number.

If the coefficients an of the powers series ∑n an un are algebraic numbers of the assumed extension or ordinary rationals, the power series in s converges for s=O(p) under rather mild conditioons. For instance, the coefficient of the zeroth order term is 1/2. The coefficient of first order term in u is -(1/2)∑k (sks*k) -1= -2∑k (1+4yk2)-1.

If this sum converges to a well-defined rational if ξ is analogous to a rational polynomial. Same should happen also for the higher coefficients cn. For large values of n the coefficients cn should become integers to guarantee the convergence of the sum for all primes p. A more general condition would be that the sums defining the coefficients give algebraic integers or even transcendentals defining a finite-D extension of rationals with p-adic norm not larger than 1 for the extension defined by zeta.

4. Algebraic integers are roots of monic polynomials with integer coefficients such that the coefficient of the leading order term is equal to 1. By multiplying ξ by the product Y=∏ sks*k one obtains an expression, which is formally a monic polynomial if one can regard Y as a p-adic integer, which is p-adically finite for every p. Formally this operation does not affect the values of the roots.
One can deduce formal expressions for the Taylor coefficiens of ξ(s).
1. Taking u=s(s-1) to be the variable, the coefficients of un in ξ(s)= ξ1(u) are given by

Unk∈ Un Xk-1 , Xk=sks*k .

2. The calculation of the coefficients cn is simple. In particular, c1 and c2 can written as

c1= (1/2)∑iXi-1 ,

c2= (1/2)∑i≠ jXi-1Xj

=(1/2)∑i,jXi-1Xj -(1/2) ∑iXi-2

= (1/2)c12 - ∑iXi-2 .

The calculation reduces to the calculation of sums ∑iXi-k, k=1,2.

3. Also the higher coefficients cn can be calculated in a similar way recursively by subtracting from the sum ∑i1...inik Xi1-1= c1n without the constraint pi≠ pj≠ ... the sums for which 2,3,...,n primes are identical. One obtains a sum over all partitions of Un. A given partition {i1, ..., ik} contributes to the sum the term

di1, ... , ikl=1k cil ,

i=1k ni=n .

The coefficient di1, ..., ik tells the number of different partitions with same numbers i1, ..., ik of elements, such that the ni elements of the subset correspond to the same prime so that this subset gives cni. Note that the same value of i can appear several times in {i1, ..., ik}.

The outcome is that the expressions of cn reduce to the calculation of the numbers Ak=∑i Xi-k.

Could one deduce conditions on the coefficients of ξ from number theoretical democracy?

Can one pose additional conditions in the case of ζ or ξ? I have difficulties in avoiding a tendency to bring in some number theoretic mysticism in hope say something interesting of the values of the coefficient Xn in the power series ξ= cnun, u=s(s-1), which can calculated from the Hadamard product representation. Number theoretical democracy between p-adic number fields defines one form of mysticism.

There is however also a real problem involved. There is a highly non-trivial problem involved. One can estimate the real coefficients Xk only as a rational approximation since infinite sums of powers of 1/Xk are involved. The p-adic norm of the approximation is very sensitive to the approximation.

Therefore it seems that one must pose additional conditions and the conditions should be such that the coefficients are mapped to numbers in extension of p-adic numbers by the inverse of I as such so that they should be algebraic numbers or even transcendentals in a finite-D transcendental extension of rationals, if such exists.

1. One could argue that the coefficients cn must obey a number theoretical democracy, which would mean that they can distinguish p-adically only between the set of primes pk appearing as divisors of n and the remaining primes. One could require that cn is a number in a finite-D extension of rationals involving only rational primes dividing n.
2. One could pose an even stronger condition: the coefficients cn must belong to an n-D algebraic extension of rationals and thus be determined by a polynomial of degree n. Polynomials P of rational coefficients pn bring in failure of the number theoretic democracy unless one has pn∈ {0,+/- 1}. For p=2 one does not obtain algebraic numbers. For p=3 this would bring in 51/2.
3. These conditions would guarantee that for a given prime p the coefficients of the expansion would be unaffected by the canonical identification I and at the limit p→ &infty; the Taylor coefficients of p-adic ξp would be identical with those of ξ.
4. One could allow finite-D transcendental extensions of p-adic numbers. These exist. Since ep is an ordinary p-adic number, there is an infinite number of extensions with a basis given by the powers roots ek/n, k=1,..., np-1 define a finite-D transcendental extension of p-adics for every prime p.

The strongest hypothesis is that the coefficients ck are expressible solely as polynomials of this kind of extensions with coefficients, which are algebraic numbers of integers in an extension of rationals by a k:th order polynomial Pk, whose coefficients belong to {0,+/- 1}.

This picture suggests a connection with the hyperbolic geometry H2 of the upper half-plane, which is associated with ζ and ξ via Langlands correspondence.
1. The simplest option is that the roots of Pk correspond to the k:th roots xi of unity satisfying xik=1 so that cos(n2π/k) and sin(n2π/k) would appear as coefficients in the expression of ck. The numbers ek/n would be hyperbolic counterparts for the roots of unity.
2. The coefficients ck would be of form

ck= ∑i,jck,ijei/k e(-1) 1/2 2π (j/n) ,

ck,rs ∈ {0,+/- 1} .

The coefficients could be seen as Mellin-Fourier transforms of functions defined in a discretized hyperbolic space H2 defined by 2-D mass shell with coordinates (cosh(η), sinh(η)cos(phi), sinh(η)sin(φ)), η = i/k, φ= 2π j/n. η is the hyperbolic angle defining the Lorentz boost to get the momentum from rest momentum and φ defines the direction of space-like part of the momentum. Upper complex plane defines another representation of H2. The values of functions are in the set {0,+/- 1}.

3. The points of H2 associated with a particular ck would correspond to the orbit of a discrete subgroup of the Lorentz group SO(1,1)× SO(2)⊂ SO(1,2) ⊂ SL(2,R) ( SL(2,R) is the covering of SO(1,2)).

A good guess is that this discretization could be regarded as a tessellation of H2 and whether other tessellations (there exists an infinite number of them corresponding to discrete subgroups of SL(2,R) could be associated with other L-functions. Riemann zeta is related by Mellin transform to Jacobi theta function (see this) so that SL(2,C), having SL(2,R) as subgroup acting as isometries of H2, is the appropriate group.

4. The points of H2 associated with a particular ck would correspond to the orbit of a discrete subgroup of SO(1,1)× SO(2)⊂ SO(1,2) ⊂ SL(2,R) (SL(2,R) is the covering of SO(1,2)).

A good guess is that this discretization could be regarded as a tessellation of H2 and whether other tessellations (there exists an infinite number of them corresponding to discrete subgroups of SL(2,R) could be associated with other L-functions. Mellin transform relates Jacobi theta function (see this), which is a modular form, to 2ξ/s(s-1). Therefore SL(2,C), having SL(2,R) as subgroup acting as isometries of H2, is the appropriate group.

Note that the modular forms associated with the representations of algebraic subgroups of SL(2,C) defined by finite algebraic extensions of rationals correspond to L-functions analogous to ζ. Now one would have a hyperbolic extension of rationals inducing a finite-D extension of p-adic numbers.

Just for curiosity and to see how the proposal could fail, one can look at what happens for the first coefficient c1 in ξ(s)= ξ1(s(s-1))= ∑ cnsn.
1. c1 would be exceptional since it cannot depend on any prime. c2 could involve only p=2, and so on.
2. The only way out of the problem is to allow finite-D transcendental extensions of p-adic numbers. These exist. Since ep is an ordinary p-adic number, there is an infinite number of extensions with a basis given by the powers roots ek/n, k=1,..., np-1 define a finite-D transcendental extension of p-adics for every prime p. For ξ the extension by roots of unity could be infinite-dimensional.

The roots ek/n, k∈ {1, ..., n} belong to this extension for all primes p and are in this sense universal. One can construct from the powers of ek/n expressions for c1 as c1=∑k ake-k/n, ak ∈{ +/- =0,+/- 1}.

3. This would allow to get estimates for n using x1=dξ/ds(0)∼ .011547854 =2c1 as an input:

c1=∑ ake-k/n=x1/2 .

For instance, the approximation cn= e1- e(n-1)/n would give a rough starting point approximation n ∼ 117. It is of course far from clear whether a reasonably finite value of n can reproduce the approximate value of c1.

See the article Some New Ideas Related to Langlands Program viz. TGD or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.