Some weeks ago, the Riemann zeta function and the Riemann hypothesis were again in the news. Sir Michael Atiyah proposed a (wrong) failed proof of the Riemann hypothesis. My blog has an history of being controversial from time to time. … Continue reading
Today, I return to my best friends. The polylogs! Are you ready for polylog wars? The polylogarithm can be represented by the next integral: (1) As you know, if you follow my blog, you have … Continue reading
Hi, there! We are going to explore more mathematical objects in this post. Today, the objects to study are theta functions. A prototype is the Jacobi theta function: (1) where and . It satisfies the functional equation (2) … Continue reading
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
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
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
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
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
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