By David H. Bailey, Visit Amazon's Jonathan M. Borwein Page, search results, Learn about Author Central, Jonathan M. Borwein, , Neil J. Calkin, Roland Girgensohn, D. Russell Luke, Victor Moll

The final 20 years were witness to a basic shift within the method arithmetic is practiced. With the ongoing boost of computing energy and accessibility, the view that genuine mathematicians do not compute not has any traction for a more moderen iteration of mathematicians which could rather reap the benefits of computer-aided study, in particular given the scope and availability of contemporary computational programs resembling Maple, Mathematica, and MATLAB. The authors supply a coherent number of available examples of recent arithmetic matters within which clever computing performs an important function.

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I believe that elementary number theory and the rest of mathematics should be pursued more in the spirit of experimental science, and that you should be willing to adopt new principles... And the Riemann Hypothesis isn’t self-evident either, but it’s very useful. A physicist would say that there is ample experimental evidence for the Riemann Hypothesis and would go ahead and take it as a working assumption. · · · We may want to introduce it formally into our mathematical system (Greg Chaitin, [44, p.

For the time being, we simply note that both Mathematica and Maple have incorporated some reasonably good numerical facilities for this purpose, and it is often sufficient to rely on these packages when numerical values are needed. ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 5. Computation of Multivariate Zeta Constants 43 5 Computation of Multivariate Zeta Constants One class of mathematical constants that has been of particular interest to experimental mathematicians in the past few years are multivariate zeta constants.

48 Chapter 2. 28). More study, however, will be needed to understand this phenomenon. 28) have no “explanation,” there are tantalizing echoes of the elliptic theory described in [46] that explains the series for 1/π as we now partially reprise. We first define the theta functions θ3 , θ4 and θ2 ∞ ∞ ∞ 2 qn , θ3 (q) := 2 (−1)n q n , θ4 (q) := n=−∞ n=−∞ n=−∞ 2 q (n+1/2) , θ2 (q) := for |q| < 1. We next identify the invariant θ22 −π√N e . θ32 √ We denote the complementary modulus k := 1 − k 2 in terms of which it transpires that Jacobi’s identity θ34 = θ44 + θ24 (see [46]) implies kN = kN = θ22 −π√N .