Surprising Sinc Sums and Integrals

Abstract

His main area of research has been classical analysis, particularly summability theory.

Exploratory Experimentation: Digitally-Assisted Discovery and Proof

“…Consider the following empiricallydiscovered identity where the denumerators range over the primes. Provably, the following is true: The analogous "sum equals integral" identity remains valid for more than the first 10, 176 primes but stops holding after some larger prime, and thereafter the 'sum minus integral" is positive but much less than one part in a googolplex [3]. It is hard to imagine that inductive mathematics alone will ever be able to handle such behaviour .…”

“…This is equally true of extreme-precision calculation -a prerequisite for much of my own work [3,7] and others [9] -or in combinatorics.…”

“…3 • Web Databases, including Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, Planet Math, Digital Library of Mathematical Functions (DLMF), MacTutor, Amazon, and many more sources that are not always viewed as part of the palette. Nor is necessary that one approve unreservedly, say of the historical reliability of MacTutor, to acknowledge that with appropriate discrimination in its use it is a very fine resource.…”

Computer-Assisted Discovery and Proof

“…As a single instance, it was discovered by Robert Baillie [2], after reading [21, §2.5], that for N = 1, 2, 3, 4, 5, and 6 (but not 7), …”

“…A:=proc() local N; N:=args [1]; if nargs>1 then Digits:=args [2] fi; identify(evalf(G(N))), identify(evalf(a0(N)))+'n'*identify(evalf(a1(N))) end: B:=proc() local N; N:=args [1]; if nargs>1 then Digits:=args [2] fi; identify(evalf(g(N))), identify(evalf(b0(N)))+'n'*identify(evalf(b1(N))) end:…”