### Generalization of Riemann zeta to Dedekind zeta and adelic physics

A further insight to adelic physics comes from the possible physical interpretation of the L-functions appearing also in Langlands program (see this. The most important L-function would be generalization of Riemann zeta to extension of rationals. I have proposed several roles for ζ, which would be the simplest L-function assignable to rational primes, and for its zeros.

- Riemann zeta itself could be identifiable as an analog of partition function for a system with energies given by logarithms of prime. In ZEO this function could be regarded as complex square root of thermodynamical partition function in accordance with the interpretation of quantum theory as complex square root of thermodynamics.

- The zeros of zeta could define the conformal weights for the generators of super-symplectic algebra so that the number of generators would be infinite. The rough idea - certainly not correct as such except at the limit of infinitely large CD - is that corresponding functions would correspond to functions of radial light-like coordinate r
_{M}of light-cone boundary (boundary of causal diamond) of form (r_{M}/r_{0})^{sn}, s_{n}=1/2+iy, s_{n}would be radial conformal weight. Periodic boundary conditions for CD do not allow all possible zeros as conformal weights so that for given CD only finite subset corresponds to generators of supersymplectic algebra. Conformal confinement would hold true in the sense that the sum s_{n}for physical states would be integer. Roots and their conjugates should appear as pairs in physical states.

- On basis of numerical evidence Dyson (see this) has conjectured that the Fourier transform for the set formed by zeros of zeta consists of primes so that one could regard zeros as one-dimensional quasi-crystal. This hypothesis makes sense if the zeros of zeta decompose into disjoint sets such that each set corresponds to its own prime (and its powers) and one has p
^{iy= Um/n=exp(i2π m/n) (see the appendix of this). This hypothesis is motivated by number theoretical universality. } - I have considered the possibility (see this) that the inverse of the electro-weak U(1) coupling constant for a gauge field assignable to the Kähler form of CP
_{2}corresponds to poles of the fermionic zeta ζ_{F}(s)= ζ(s)/ζ(2s) coming from s_{n}/2 (denominator) and pole at s=1 (numerator) zeros of zeta assignable to rational primes. Here one can consider scaling of argument of ζ_{F}(s). More general coupling constant evolutions could correspond to ζ_{F}(m(s)) where m(s)= (as+b)/(cs+d) is Möbius transformation performed for the argument mapping upper complex plane to itself so that a,b,c,d are real and also rational by number theoretical universality.

- The definition of Dedekind zeta function ζ
_{K}relies on the product representation and analytic continuation allows to deduce ζ_{K}elsewhere. One has a product over prime ideals of K/Q of rationals with the factors 1/(1-p^{-s}) associated with the ordinary primes in Riemann zeta replaced with the factors X(P) =1/(1-N_{K/Q}(P)^{-s}), where P is prime for the integers O(K) of extension and N_{K/Q}(P) is the norm of P in the extension. In the region s>1 where the product converges, ζ_{K}is non-vanishing and s=1 is a pole of ζ_{K}. The functional identifies of ζ hold true for ζ_{K}as well. Riemann hypothesis is generalized for ζ_{K}.

- It is possible to interpret ζ
_{K}in terms of a physical picture. By the general results (see this) one N_{K/Q}(P)= p^{r}, r>0 integer. One can deduce for r a general expression. This implies that one can arrange in ζ_{K}all primes P for which the norm is power or given p in the same group. The prime ideals p of ordinary integers decompose to products of prime ideals P of the extension: one has p= ∏_{r=1}^{g}P_{r}^{er}, where e_{r}is so called ramification index. One can say that each factor of ζ decomposes to a product of factors associated with corresponding primes P with norm a power of p. In the language of physics, the particle state represented by p decomposes in an improved resolution to a product of many-particle states consisting of e_{r}particles in states P_{r}, very much like hadron decomposes to quarks.

The norms of N

_{K/Q}(P_{r}) = p^{dr}satisfy the condition ∑_{r=1}^{g}d_{r}e_{r}= n. Mathematician would say that the prime ideals of Q modulo p decompose in n-dimensional extension K to products of prime power ideals P_{r}^{er}and that P_{r}corresponds to a finite field G(p,d_{r}) with algebraic dimension d_{r}. The formula ∑_{r=1}^{g}d_{r}e_{r}= n reflects the fact the dimension n of extension is same independent of p even when one has g<n and ramification occurs.

Physicist would say that the number of degrees of freedom is n and is preserved although one has only g<n different particle types with e

_{r}particles having d_{r}internal degrees of freedom. The factor replacing 1/(1-p^{-s}) for the general prime p is given by ∏_{r=1}^{g}1/(1-p^{-erdrs}).

- There are only finite number of ramified primes p having e
_{r}>1 for some r and they correspond to primes dividing the so called discriminant D of the irreducible polynomial P defining the extension. D mod p obviously vanishes if D is divisible by p. For second order polynomials P=x^{2}+bx+c equals to the familiar D=b^{2}-4c and in this case the two roots indeed co-incide. For quadratic extensions with D= b^{2}-4c>0 the ramified primes divide D.

**Remark:**Resultant R(P,Q) and discriminant D(P)= R(P,dP/dx) are elegant tools used by number theorists to study extensions of rationals defined by irreducible polynomials. From Wikipedia articles one finds an elegant proof for the facts that R(P,Q) is proportional to the product of differences of the roots of P and Q, and D to the product of squares for the differences of distinct roots. R(P,Q)=0 tells that two polynomials have a common root. D mod p=0 tells that polynomial and its derivative have a common root so that there is a degenerate root modulo p and the prime is indeed ramified. For modulo p reduction of P the vanishing of D(P) mod p follows if D is divisible by p. There exist clearly only a finite number of primes of this kind.

Most primes are unramified. If one has maximum number n of factors in the decomposition and e

_{r}=1, maximum splitting of p occurs. The factor 1/(1-p^{-s}) is replaced with its n:th power 1/(1-p^{-s})^{n}. The geometric interpretation is that space-time sheet is replaced with n-fold covering and each sheet gives one factor in the power. It is also possible to have a situation in which no splitting occurs and one as e_{r}=1 for one prime P_{r}=p. The factor is in this case equal to 1/(1-p^{-ns}).

_{L}/ζ

_{K}is so called Artin L-function of the regular representation (group algebra) of Galois group factorizing in terms of irreps of Gal(L/K) is

*holomorphic*(no poles!) so that ζ

_{L}must have also the zeros of ζ

_{K}. This holds in the special case K=Q. Therefore extension of rationals can only bring new zeros but no new poles!

- This result is quite far reaching if one accepts the hypothesis about super-symplectic conformal weights as zeros of ζ
_{K}and the conjecture about coupling constant evolution. In the case of ζ_{F,K}this means new poles meaning new conformal weights due to increased complexity and a modification of the conjecture for the coupling constant evolution due to new primes in extension. The outcome looks physically sensible.

- Quadratic field Q(m
^{1/2}) serves as example. Quite generally, the factorization of rational primes to the primes of extension corresponds to the factorization of the minimal polynomial for the generating element θ for the integers of extension and one has p= P_{i}^{ei}, where e_{i}is ramification index. The norm of p factorizes to the produce of norms of P_{i}^{ei}.

Rational prime can either remain prime in which case x^{2}-m does not factorize mod p, split when x^{2}-m factorizes mod P, or ramify when it divides the discriminant of x^{2}-m = 4m. From this it is clear that for unramfied primes the factors in ζ are replaced by either 1/(1-p^{-s})^{2}or 1/(1-p^{-2s})= 1/(1-p^{-s})(1+p^{-s}). For a finite number of unramified primes one can have something different.

For Gaussian primes with m=-1 one has e

_{r}=1 for p mod 4=3 and e_{r}=2 for p=~mod~4=1. z_{K}therefore decomposes into two factors corresponding to primes p ~mod~4=3 and p mod 4=1. One can extract out Riemann zeta and the remaining factor

∏

_{p mod 4=3}1/(1-p^{-s}) × ∏_{p mod 4=1}1/(1+p^{-s})

should be holomorphic and without poles but having possibly additional zeros at critical line. That ζ

_{K}should possess also the poles of ζ as poles looks therefore highly non-trivial.

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