LOG#167. D-dimensional laws(II).

Let me begin this article in D=d+1 spacetime. We are going to study quantum gases and their statistics in multidimensional space. Usual notation:     In D=d+1 spacetime, the massless free ideal relativistic gas satisfies, as we will show, certain … Continue reading

LOG#162. Polylogia flashes(IV).

In this final post (by the moment) in the polylogia series we will write some additional formulae for polylogs and associated series. Firstly, we have (1)   and now, if (2)   (3)   The next identity also holds   … Continue reading

LOG#161. Polylogia flashes(III).

In the third post of this series I will write more fantastic identities related to our friends, the polylogs! (1)   and by analytic continuation that equation can be extended to all . In fact (2)   such as , … Continue reading

LOG#160. Polylogia flashes(II).

The polylogarithm or Jonquière’s function is generally defined as     Do not confuse with the logarithm integral in number theory, which is     such as     and     . In fact, notation can be confusing sometimes … Continue reading

LOG#159. Polylogia flashes (I).

In the next series of post, I am going to define (again) and write some cool identities of several objects that mathematicians and physicists know as zeta functions and polylogarithms. I have talked about them already here http://www.thespectrumofriemannium.com/2012/11/07/log051-zeta-zoology/ 1. We … Continue reading

LOG#154. Moonshine and 42: THE PAPER.

“(…) The answer to the Great Question…is…Forty-two(…)” said Deep Thought with infinity majesty and calm (frequently found quote from The Hitchhiker’s Guide to the Galaxy, Douglas Adams, London 1979) Dear readers, yesterday, while I was editing, re-editing and writing my … Continue reading

LOG#104. Primorial objects.

My post today will be discussing two ideas: the primorial and the paper “The product over all primes is “ (2003). The primorial is certain generalization of the factorial, but running on prime numbers. While the factorial is defined as … Continue reading

LOG#079. Zeta multiple integral.

My second post this day is a beautiful relationship between the Riemann zeta function, the unit hypercube and certain multiple integral involving a “logarithmic and weighted geometric mean”. I discovered it in my rival blog, here: http://tardigrados.wordpress.com/2013/01/08/la-funcion-zeta-de-riemann-definida-en-terminos-de-integrales-multiples/ First of all, … Continue reading

LOG#051. Zeta Zoology.

This log-entry is an exploration journey… To boldly go, where no zeta function has gone before… Riemann zeta function The Riemann zeta function is an object related to prime numbers. In general, it is a function of complex variable defined … Continue reading

LOG#050. Why riemannium?

TABLE OF CONTENTS DEDICATORY 1. THE RIEMANN ZETA FUNCTION ζ(s) 2. THE RIEMANN HYPOTHESIS 3. THE HILBERT-POLYA CONJECTURE 4. RANDOM MATRIX THEORY 5. QUANTUM CHAOS AND RIEMANN DYNAMICS 6. THE SPECTRUM OF RIEMANNIUM 7. ζ(s) AND RENORMALIZATION 8. ζ(s) AND … Continue reading