Ulla mentioned in the comment section of the earlier posting an intervew of Matthew Watkins. The pages of Matthew Watkins about all imaginable topics related to Riemann zeta are excellent and I can only warmly recommend. I was actually in contact with him for years ago and there might be also TGD inspired proposal for strategy proving Riemann hypothesis at the pages of Matthew Watkins.
The interview was very inspiring reading. MW has very profound vision about what mathematics is and he is able to express it in understandable manner. MW tells also about the recent work of Connes applying p-adics and adeles(!) to the problem. I would guess that these are old ideas and I have myself speculated about the connection with p-adics for long time ago.
MW tells in the interview about the thermodynamical interpretation of zeta function. Zeta reduces to a product ζ(s)= ∏pZp(s) of partition functions Zp(s)=1/[1-p-s] over particles labelled by primes p. This relates very closely also to infinite primes and one can talk about Riemann gas with particle momenta/energies given by log(p). s is in general complex number and for the zeros of the zeta one has s=1/2+iy. The imaginary part y is non-rational number. At s=1 zeta diverges and for Re(s)≤1 the definition of zeta as product fails. Physicist would interpret this as a phase transition taking place at the critical line s=1 so that one cannot anymore talk about Riemann gas. Should one talk about Riemann liquid? Or - anticipating what follows- about quantum liquid? What the vanishing of zeta could mean physically? Certainly the thermodynamical interpretation as sum of something interpretable as thermodynamical probabilities apart from normalization fails.
The basic problem with this interpretation is that it is only formal since the temperature parameter is complex. How could one overcome this problem?
A possible answer emerged as I read the interview.
- One could interpret zeta function in the framework of TGD - or rather in zero energy ontology (ZEO) - in terms of square root of thermodynamics! This would make possible the complex analog of temperature. Thermodynamical probabilities would be replaced with probability amplitudes.
- Thermodynamical probabilities would be replaced with complex probability amplitudes, and Riemann zeta would be the analog of vacuum functional of TGD which is product of exponent of Kähler function - Kähler action for Euclidian regions of space-time surface - and exponent of imaginary Kähler action coming from Minkowskian regions of space-time surface and defining Morse function.
In QFT picture taking into account only the Minkowskian regions of space-time would have only the exponent of this Morse function: the problem is that path integral does not exist mathematically. In thermodynamics picture taking into account only the Euclidian regions of space-time one would only the exponent of Kähler function and would lose interference effects fundamental for QFT type systems.
In quantum TGD both Kähler and Morse are present. With rather general assumptions the imaginary part and real part of exponent of vacuum functional are proportional to each other and to sum over the values of Chern-Simons action for 3-D wormhole throats and for space-like 3-surfaces at the ends of CD. This is non-trivial.
- Zeros of zeta would in this case correspond to a situation in which the integral of the vacuum functional over the "world of classical worlds" (WCW) vanishes. The pole of ζ at s=1 would correspond to divergence fo the integral for the modulus squared of Kähler function.
What the vanishing of the zeta could mean if one accepts the interpretation quantum theory as a square root of thermodynamics?
- What could the infinite value of zeta at s=1 mean? The The interpretation in terms of square root of thermodynamics implied following. In zero energy ontology zeta function function decomposition to ∏p Zp(s) corresponds to a product of single particle partition functions for which one can assigns probabilities p-s/Zp(s) to single particle states. This does not make sense physically for complex values of s.
- In ZEO one can however assume that the complex number p-sn define the entanglement coefficients for positive and negative energy states with energies nlog(p) and -nlog(p): n bosons with energy log(p) just as for black body radiation. The sum over amplitudes over over all combinations of these states with some bosons labelled by primes p gives Riemann zeta which vanishes at critical line if RH holds.
- One can also look for the values of thermodynamical probabilities given by |p-ns|2= p-n at critical line. The sum over these gives for given p the factor p/(p-1) and the product of all these factors gives ζ (1)=∞. Thermodynamical partition function diverges. The physical interpretation is in terms of Bose-Einstein condensation.
- What the vanishing of the trace for the matrix coding for zeros of zeta defined by the amplitudes is physically analogous to the statement ∫ Ψ dV=0 and is indeed true for many systems such as hydrogen atom. But what this means? Does it say that the zero energy state is orthogonal to vacuum state defined by unit matrix between positive and negative energy states? In any case, zeros and the pole of zeta would be aspects of one and same thing in this interpretation. This is an something genuinely new and an encouraging sign. Note that in TGD based proposal for a strategy for proving Riemann hypothesis, similar condition states that coherent state is orthogonal to a "false" tachyonic vacuum.
- RH would state in this framework that all zeros of ζ correspond to zero energy states for which the thermodynamical partition function diverges. Another manner to say this is that the system is critical. (Maximal) Quantum Criticality is indeed the key postulate about TGD Universe and fixes the Kähler coupling strength characterizing the theory uniquely (plus possible other free parameters). Quantum Criticality guarantees that the Universe is maximally complex. Physics as generalized number theory would suggest that also number theory is quantum critical! When the sum over numbers proportional to propabilities diverges, the probabilities are considerably different from zero for infinite number of states. At criticality the presence of fluctuations in all scales implying fractality indeed implies this. A more precise interpretation is in terms of Bose-Eisntein condensation.
- The postulate that all zero energy states for Riemann system are zeros of zeta and critical in the sense being non-normalizable combined with the fact that s=1 is the only pole of zeta implies that the all zeros correspond to Re(s)=1/2 so that RH follows from purely physical assumptions. The behavior at s=1 would be an essential element of the argument. Note that in ZEO coherent state property is in accordance with energy conservation. In the case of coherent states of Cooper pairs same applies to fermion number conservation.
With this interpretation the condition would state orthogonality with respect to the coherent zero energy state characterized by s=0, which has finite norm and does not represent Bose-Einstein condensation. This could give a connection with the proposal for the strategy for proving Riemann Hypothesis by replacing eigenstates of energy with coherent states and the two approaches could be unified. Note that in this approach conformal invariance for the spectrum of zeros of zeta is the axiom yielding RH and could be seen as counterpart for the fundamental role of conformal invariance in modern physics and indeed very natural in the vision about physics as generalized number theory.
For background see the chapter Riemann Hypothesis and Physics.