DBN Upper Bound is a Polymath project related to the theory of the Riemann zeta function, based at this Polymath wiki. For the background, theory and discussion on results, please head to the proposal and subsequent threads on Prof. Tao's blog. The computational part of the research is being hosted at this github project.

This Boinc project now plans to scale up the computations, and achieve tighter conditional bounds, using Internet-connected computers. You can contribute to the research by running a free program on your computer.

For a visual understanding with cool graphics and a summary of the results achieved, please check this visual guide created by Rudolph, who has also written the Arb scripts used for the scaled up computations

Computed SingleStoredSums_10e19+12010

Computed SingleStoredSums_2x10e20+66447

Computed SingleStoredSums_9x10e21+70686

Currently running jobs to establish Λ <= 0.1 conditionally (Update: 50%+ complete)

• Tasks may face errors on processors without the AVX2 instruction set, so please check your processor model and instruction set before proceeding.

• Also, please ensure virtualization is enabled within the BIOS as the tasks require the use of VirtualBox.

• Lastly, each task requires a minimum allocation of 2 GB of RAM, so please ensure Virtualbox has sufficient allocation of RAM.

There are multiple types of computations that are relevant to the project

1. **Storedsums** - One-time computation of a Taylor coefficients matrix at a certain height X.
This can then be used repeatedly to evaluate approximations to H_{t}(z)
near X using Taylor polynomials. For more details check this
Arb
or Pari/GP
code

The currently available jobs are computing this matrix for X near 2x10^20, which should likely lead to Λ <= 0.11 in subsequent steps (conditional on RH being true upto this X).

2. **Establishing a Barrier near height X** - Here we actually evaluate H_{t}(z)
approximations along a
rectangle contour mesh [(X,X+1),(y_{0},1)] for small increments in t from [0,t_{0}].
If the winding number turns out to be 0 for each such t, a barrier region is established.
The storedsums matrix is used as a helper and significantly speeds up this step.

Till now this step has been quick enough to perform on a standalone machine, but in the future even this could be processed through the grid computing setup. For more details, check this Arb or Pari/GP code

3. **Lemmabounds for an N range** - Sometimes the analytically derived lower bound
for H_{t}(z) is not positive at N_{a}=sqrt(X/(4π)) and lemmabounds have to be
calculated for each N between N_{a} and N_{b} where the analytical bound becomes positive.
We are currently working around this step by carefully choosing X, but for instance the unconditional bound
of Λ <= 0.22 required this. For more details, check this
Arb
or Pari/GP code

©2018 D.H.J. Polymath