.... "If the brain was so simple that we could understand it, then we would be so simple that we couldn't." -- Emerson M. Pugh

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Monday, January 5, 2009

Riemann Hypothesis Prediction

John Brockman's "Edge" group was asked, as their yearly question this year, what scientific development do they expect to see "change everything" within their lifetime. Clifford Pickover's answer here, is that the Riemann Hypothesis, one of math's most enduring unsolved problems, will be proven.

(Worth reading many of the other Edge respondents as well...) Anonymous said...

Re: Riemann hypothesis

http://www.mmdnewswire.com/famous-math-problem-4626.html

Sam Gilbert Anonymous said...

Shecky, I like your blog. Thanks.

Here is an excerpt from the book:

The Riemann Hypothesis & the Roots of the Riemann Zeta Function

by Samuel W. Gilbert

available from amazon.com
http://www.riemannzetafunction.com

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, where σ =½. 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…