Krzysztof Maslanka
Jagiellonian University
ul. Orla 171
30-244 Cracow Poland

The Zeta-Function of Riemann

We may - paraphrasing the famous sentence of George Orwell - say that "all mathematics is beautiful, yet some is more beautiful then the other". But the most beautiful in all mathematics is the zeta function. There is no doubt about it.

In fact, there are several functions called zeta: Riemann, Hurwitz, Epstein,... There is also another, relatively young zeta, called Hawking zeta. This one is well known in physics and in quantum cosmology. I first got acquainted with the zeta family when studying cosmology. I had no mathematical ambitions at that time. I only wanted to trace various cosmic enigmas with mathematical tools.

Suddenly everything went awry and all my cosmology decayed. Simply collapsed. No fresh theoretical ideas, no real progress in understanding. Only a flood of obserwational data. Map making, curve fitting. An endless race for money.

I was left with great emptiness. And with many pages of strange calculations. This paper below has been extracted from a pile of manuscript which was apparently meant for burning down. But it wasn't.

The paper is supposed to be a modest tribute to the memory of great Bernhard Riemann, a real genius, whose 1859 paper on zeta-function is still a source of unsolved problems and fruitful ideas. As E. C. Titchmarsh put it - "it is by no means certain that its riches are even now exhausted ".

Hypergeometric-like Representation of the Zeta-Function of Riemann
(Cracow Observatory preprint 1997/30)
main text (dvi file) and figure (postscript file)
or gzip'ed postscript file

Hypergeometric-like Representation of the Zeta-Function of Riemann.
Part II. Applications to Non-trivial Zeros of Zeta
(Cracow Observatory preprint 1997/34, available soon)