Saturday, March 28, 2009

the Riemann zeta function

is one of the most beautiful math functions I've ever seen.




via the new shelton wet/dry
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3 comments:

Sam Gilbert said...

I'm so happy you like the zeta function. I like it too. Perhaps your readers would be interested in my new book.

an excerpt from

The Riemann Hypothesis & the Roots of the Riemann Zeta Function

by Samuel W. Gilbert

www.riemannzetafunction.com

© U. S. Copyrights - 2009, 2008, 2005

This book is concerned with the geometric convergence of the Dirichlet series representation of the Riemann zeta function at its roots in the critical strip. The objectives are to understand why non-trivial roots occur in the Riemann zeta function, to define the roots mathematically, and to resolve the Riemann hypothesis.

The Dirichlet infinite series parts of the Riemann zeta function diverge everywhere in the critical strip. Therefore, it has always been assumed that the Dirichlet series representation of the zeta function is useless for characterization of the roots in the critical strip. In this work, it is shown that this assumption is completely wrong.

The Dirichlet series representation of the Riemann zeta function diverges algebraically everywhere in the critical strip. However, the Dirichlet series representation does, in fact, converge at the roots in the critical strip ̵and only at the roots in the critical strip in a special geometric sense. Although the Dirichlet series parts of the zeta function diverge both algebraically and geometrically everywhere in the critical strip, at the roots of the zeta function, the parts are geometrically equivalent and their geometric difference is identically zero.

At the roots of the Riemann zeta function, the two Dirichlet infinite series parts are coincidently divergent and are geometrically equivalent. The roots of the zeta function are the only points in the critical strip where infinite summation and infinite integration of the terms of the Dirichlet series parts are geometrically equivalent. Similarly, the roots of the zeta function with the real part of the argument reflected in the critical strip are the only points where infinite summation and infinite integration of the terms of the Dirichlet series parts with reflected argument are geometrically equivalent.

Reduced, or simplified, asymptotic expansions for the terms of the Riemann zeta function series parts at the roots, equated algebraically with reduced asymptotic expansions for the terms of the zeta function series parts with reflected argument at the roots, constrain the values of the real parts of both arguments to the critical line. Hence, the Riemann hypothesis is correct.

At the roots of the zeta function in the critical strip, the real part of the argument is the exponent, and the real and imaginary parts combine to constitute the coefficients of proportionality in geometrical constraints of the discrete partial sums of the series terms by a common, divergent envelope.

Values of the imaginary parts of the first 50 roots of the Riemann zeta function are calculated using derived formulae with 80 correct significant figures using a laptop computer. The first five imaginary parts of the roots are:

14.134725141734693790457251983562470270784257115699243175685567460149963429809256…
21.022039638771554992628479593896902777334340524902781754629520403587598586068890…
25.010857580145688763213790992562821818659549672557996672496542006745092098441644…
30.424876125859513210311897530584091320181560023715440180962146036993329389333277…
32.935061587739189690662368964074903488812715603517039009280003440784815608630551…

It is further demonstrated that the derived formulae yield calculated values of the imaginary parts of the roots of the Riemann zeta function with more than 330 correct significant figures.

continued...

March 29, 2009 at 7:58 AM
kaile said...

wow sounds fascinating!

March 29, 2009 at 11:15 PM
Sam Gilbert said...

Thanks Kaile. I realy like your design ideas.

March 31, 2009 at 6:09 AM

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