128 research outputs found

### Riemann zeta zeros and prime number spectra in quantum field theory

The Riemann hypothesis states that all nontrivial zeros of the zeta function
lie in the critical line $\Re(s)=1/2$. Hilbert and P\'olya suggested that one
possible way to prove the Riemann hypothesis is to interpret the nontrivial
zeros in the light of spectral theory. Following this approach, we discuss a
necessary condition that such a sequence of numbers should obey in order to be
associated with the spectrum of a linear differential operator of a system with
countably infinite number of degrees of freedom described by quantum field
theory. The sequence of nontrivial zeros is zeta regularizable. Then,
functional integrals associated with hypothetical systems described by
self-adjoint operators whose spectra is given by this sequence can be
constructed. However, if one considers the same situation with primes numbers,
the associated functional integral cannot be constructed, due to the fact that
the sequence of prime numbers is not zeta regularizable. Finally, we extend
this result to sequences whose asymptotic distributions are not "far away" from
the asymptotic distribution of prime numbers.Comment: Revised version, 18 page

### Fixed points in the family of convex representations of a maximal monotone operator

Any maximal monotone operator can be characterized by a convex function. The
family of such convex functions is invariant under a transformation connected
with the Fenchel-Legendre conjugation. We prove that there exist a convex
representation of the operator which is a fixed point of this conjugation.Comment: 13 pages, updated references. Submited in July 2002 to Proc. AM

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