Riemann zeta function

The interchain interaction of dipole moments collinear to the chain axes decays exponentially with the interchain distance (see Eq. (2.2.11)) and has no contribution descending by the power law y2 in Eq. (3.3.6). Summation over lattice sites and interchain distances involves the Riemann zeta-function... 

For excellent insight into Riemann and his mathematics (primarily focusing on the Riemann zeta function and the Riemann hypothesis ), see J. Derbyshire. Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (Penguin, New York, 2004).]... 

Keating, J. (1993). The Riemann zeta function and quantum chaology, in Quantum Chaos, eds. G. Casati, I. Guarneri and U. Smilansky (North-Holland, Amsterdam). 

Odlyzko, A.M. (1989). The 10 °-th zero of the Riemann zeta function and 70 million of its neighbors, Preprint, AT T Bell Laboratories. 

After extending the upper limit to infinity instead of cutting off at jc, and using the tabulated value of the pertinent Riemann zeta function (for = 3), we have... 

The function C (p) is called the Riemann zeta function. See H.B. Dwight, Tables of Elementary and Some Higher Mathematical Functions, 4th ed., Dover, New York, 1961, for tables of values of this function. 

Abstract The multiparticle distribution functions and density matrices for ideal Fermi gas system in the ground state are calculated for any spatial dimension. The results are expressed as determinant forms, in which a correlation kernel plays a vital role. The expression obtained for the one-dimensional Fermi gas is essentially equivalent to that observed for the eigenvalue distribution of random unitary matrices, and thus to that conjectured for the distribution of the non-trivial zeros of the Riemann zeta function. Their implications are discussed briefly. 

The expression for the pair (n = 2) distribution function in three (d = 3) dimension is well known [1,2]. However, the general one for any n and d is much less known. Interestingly, the distribution of Fermi particles in one d = 1) dimension has a mathematical structure similar to those found for the eigenvalues of the random matrices [3-5] and for the zeros of the Riemann zeta function [6,7], as shown below. In the following Sects. 14.2 and 14.3, explicit expressions for the pair and ternary distribution functions of the ideal Fermi gas system in any dimension are derived. We then find an expression for the n-particle distribution function as a determinant form in Sect. 14.4. Another representation for the multiparticle distribution for finite IV is given in terms of density matrix in Sect. 14.5. The explicit formula for correlation kernel which plays an essential role for the description of the multiparticle correlations in the Fermi system is derived in Sect. 14.6. The relationship with the theories for the random matrices and the Riemann zeta function is addressed in Sect. 14.7. 

Interestingly, it has been known that this type of correlation structure holds also for the distribution of zeros in the Riemann zeta function. The Riemann zeta function [12] for complex variable s is defined by... 

Titchmarsh EC, Heath-Brown DR (1986) The theory of the Riemann Zeta-function. Clarendon,... 

We can integrate the spectral radiance in terms of frequency given by Equation A.15.24 by using the integral representation of a Riemann-Zeta function... 

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