de Bruijn Newman Upper Bound

Julia code following Polymath Project 15. The original motivation was to explore the efficacy of Julia in this context.

Code covering sections 1 through 8 of the draft, EFFECTIVE APPROXIMATION OF HEAT FLOW EVOLUTION OF THE RIEMANN XI FUNCTION, AND AN UPPER BOUND FOR THE DE BRUIJN-NEWMAN CONSTANT (PDF, commit 0a0f729), has been written, tested, and mostly documented. Current functionality is exported from a submodule, DBNUpperBound.PM15a. The package remains substantially behind dbn_upper_bound where the original computational work is being done.

Arbitrary precision arithmetic is used by default. (Julia wraps the GNU Multiple Precision Arithmetic Library (GMP) and the GNU MPFR Library. Reference).

Installation

This version requires Julia 1+. Earlier versions, which require Julia 0.6+, can of course be checked out from this repo as well. (Julia 1.0.0, being a major release, in not backward compatible with 0.x.x releases.) To install this version and its dependencies:

Note that the SpecialFunctions dependency (for gamma and Riemann's zeta) requires both gcc and gcc-fortran.

julia> Pkg.clone("https://github.com/WilCrofter/DBNUpperBound.jl.git")

To remain current:

julia> Pkg.update("DBNUpperBound")

All unit tests pass:

$ julia test/runtests.jl
[ Info: Testing PM15a
[ Info: Testing introduction.jl
[ Info: Testing applying the fundamental solution to the heat equation.
[ Info: Testing elementary estimates
[ Info: Testing estimates for large x
[ Info: Testing new upper bound
[ Info: Testing PM15b
[ Info: Testing estimating sums
[ Info: DBNUpperBound passes tests.