The Classical Gases in the Tsallis Statistics Using the Generalized Riemann Zeta Functions

In this paper, several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation of the important results of Katsurada and Matsumoto on the mean square of the Hurwitz zeta function. Also, a relation derived here provides the starting point of a novel approach which, in a series of companion papers, yields a formal proof of the Lindelöf hypothesis. Some of the above relations motivate the need for analysing the large α behaviour of the modified Hurwitz zeta function ζ 1 ( s , α ) , s ∈ C , α ∈ ( 0 , ∞ ) , which is also presented here.

Download Full-text

Analysis and combinatorics of partition zeta functions

International Journal of Number Theory ◽

10.1142/s1793042120400023 ◽

2020 ◽

pp. 1-10

Author(s):

Robert Schneider ◽

Andrew V. Sills

Keyword(s):

Zeta Function ◽

Explicit Formula ◽

Zeta Functions ◽

Complete Information ◽

Combinatorial Proof ◽

Fixed Length ◽

The Family ◽

The Riemann Zeta Function ◽

Riemann Zeta ◽

Partial Fraction Decomposition

We examine “partition zeta functions” analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties — those summed over partitions of fixed length — which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon’s partial fraction decomposition of the generating function for partitions of fixed length.

Download Full-text

The Generalized Riemann Zeta Functions and their Applications in the Calculations of Neutron Cross Sections

Mathematics of Computation ◽

10.2307/2003975 ◽

1965 ◽

Vol 19 (92) ◽

pp. 692

Author(s):

J. W. W. ◽

M. Atoji ◽

F. L. Clark

Keyword(s):

Cross Sections ◽

Zeta Functions ◽

Riemann Zeta ◽

Neutron Cross Sections

Download Full-text

HILBERT–PÓLYA CONJECTURE, ZETA FUNCTIONS AND BOSONIC QUANTUM FIELD THEORIES

International Journal of Modern Physics A ◽

10.1142/s0217751x13500723 ◽

2013 ◽

Vol 28 (17) ◽

pp. 1350072 ◽

Cited By ~ 6

Author(s):

JULIO C. ANDRADE

Keyword(s):

Adjoint Operator ◽

Zeta Functions ◽

Field Theories ◽

Quantum Field Theories ◽

Quantum Field ◽

Functional Integrals ◽

Generating Functional ◽

Nontrivial Zeros ◽

Riemann Zeta ◽

Adjoint Operators

The original Hilbert and Pólya conjecture is the assertion that the nontrivial zeros of the Riemann zeta function can be the spectrum of a self-adjoint operator. So far no such operator was found. However, the suggestion of Hilbert and Pólya, in the context of spectral theory, can be extended to approach other problems and so it is natural to ask if there is a quantum mechanical system related to other sequences of numbers which are originated and motivated by Number Theory. In this paper, we show that the functional integrals associated with a hypothetical class of physical systems described by self-adjoint operators associated with bosonic fields whose spectra is given by three different sequence of numbers cannot be constructed. The common feature of the sequence of numbers considered here, which causes the impossibility of zeta regularizations, is that the various Dirichlet series attached to such sequences — such as those which are sums over "primes" of ( norm P)-s have a natural boundary, i.e. they cannot be continued beyond the line Re (s) = 0. The main argument is that once the regularized determinant of a Laplacian is meromorphic in s, it follows that the series considered above cannot be a regularized determinant. In other words, we show that the generating functional of connected Schwinger functions of the associated quantum field theories cannot be constructed.

Download Full-text