The Military Effectiveness of the US Armed Forces, 1919–39

Spector, Ronald H.

doi:10.1017/cbo9780511762963.005

Cited by 5 publications

(6 citation statements)

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“…Using the Möbius function it is possible to show that the partition function of the fermionic system is given by ζ(βω)/ζ(2βω) 3,6 . As discussed in the literature, from the partition function associated to the hamiltonians H B and H F we get…”

Section: The Partition Function Of the Riemann Gas And The Averagmentioning

confidence: 99%

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Thermodynamics of the bosonic randomized Riemann gas

Dueñas¹,

Svaiter²

2015

J. Phys. A: Math. Theor.

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The partition function of a bosonic Riemann gas is given by the Riemann zeta function. We assume that the hamiltonian of this gas at a given temperature β −1 has a random variable ω with a given probability distribution over an ensemble of hamiltonians. We study the average free energy density and average mean energy density of this arithmetic gas in the complex β-plane. Assuming that the ensemble is made by an enumerable infinite set of copies, there is a critical temperature where the average free energy density diverges due to the pole of the Riemann zeta function. Considering an ensemble of non-enumerable set of copies, the average free energy density is non-singular for all temperatures, but acquires complex values in the critical region.Next, we study the mean energy density of the system which depends strongly on the distribution of the non-trivial zeros of the Riemann zeta function. Using a regularization procedure we prove that the this quantity is continuous and bounded for finite temperatures.

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Section: The Partition Function Of the Riemann Gas And The Averagmentioning

confidence: 99%

“…Arithmetic quantum theory has been developed to establish connections between number theory and quantum field theory [1][2][3][4][5][6][7] . A bosonic Riemann gas is a second quantized mechanical system at temperature β −1 , with partition function given by the Riemann zeta function 8,9 .…”

Section: Introductionmentioning

confidence: 99%

Thermodynamics of the bosonic randomized Riemann gas

Dueñas¹,

Svaiter²

2015

J. Phys. A: Math. Theor.

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“…One example of such system is known as the bosonic Riemann gas. 7,8 Recently, it was presented a generalization of such a model by assuming an ensemble of Hamiltonians of arithmetic bosonic gas with some probability distribution. 9 The singularity structure for the average free energy density of this arithmetic gas in the complex β plane was discussed.…”

Section: Introductionmentioning

confidence: 99%

One-loop effective action and the Riemann zeros

Dueñas

Svaiter

Menezes

2014

Int. J. Mod. Phys. A

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We present a remarkable connection between the asymptotic behavior of the Riemann zeros and one-loop effective action in Euclidean scalar field theory. We show that in a two-dimensional space, the asymptotic behavior of the Fourier transform of two-point correlation functions fits the asymptotic distribution of the non-trivial zeros of the Riemann zeta function. We work out an explicit example, namely the non-linear sigma model in the leading order in 1/N expansion.

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“…The statistical properties of a Fermi gas whose single-particle energy levels are given by these zeros were investigated in 26,27 . Also, in order to solve questions related to the number theory using statistical-mechanics methods, some authors introduced number theory and prime numbers in quantum field theory [28][29][30][31][32][33] .…”

Section: Introductionmentioning

confidence: 99%

Riemann Zeta Zeros and Prime Number Spectra in Quantum Field Theory

Menezes

Svaiter

2013

Int. J. Mod. Phys. A

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The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line Re(s) = 1/2. Hilbert and Pólya 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.

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The Military Effectiveness of the US Armed Forces, 1919–39

Cited by 5 publications

References 0 publications

Thermodynamics of the bosonic randomized Riemann gas

Thermodynamics of the bosonic randomized Riemann gas

One-loop effective action and the Riemann zeros

Riemann Zeta Zeros and Prime Number Spectra in Quantum Field Theory

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