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Riemann Hypothesis not proved

Here’s a tweet from Alex Bellos this morning:

He’s right to be surprised – as reported in Vanguard, a Nigerian newspaper:

The 156-year old Riemann Hypothesis, one of the most important problems in Mathematics, has been successfully resolved by Nigeria Scholar, Dr. Opeyemi Enoch.

Suspicion levels are raised, as the paper also reports:

Three of the [Clay Millenium Prize] problems had been solved and the prizes given to the winners. This makes it the fourth to be solved of all the seven problems.

Unless we missed something, that’s not massively true – the only Millennium Prize problem solved so far is the Poincaré conjecture.

Terence Tao has solved the Erdős discrepancy problem!

Terence Tao has just uploaded a preprint to the arXiv with a claimed proof of the Erdős discrepancy problem.

Small gaps between large gaps between primes results

The big news last year was the quest to find a lower bound for the gap between pairs of large primes, started by Yitang Zhang and carried on chiefly by Terry Tao and the fresh-faced James Maynard.

Now that progress on the twin prime conjecture has slowed down, they’ve both turned their attentions toward the opposite question: what’s the biggest gap between subsequent small primes?

Erdős’s discrepancy problem now less of a problem

Boris Konev and Alexei Lisitsa of the University of Liverpool have been looking at sequences of $+1$s and $-1$s, and have shown using an SAT-solver-based proof that every sequence of $1161$ or more elements has a subsequence which sums to at least $2$. This extends the existing long-known result that every such sequence of $12$ or more elements has a subsequence which sums to at least $1$, and constitutes a proof of Erdős’s discrepancy problem for $C \leq 2$.

The neuroscience of mathematical beauty, or, Equation beauty contest!

Neuroscientists Semir Zeki and John Paul Romaya have put mathematicians in an MRI scanner and shown them equations, in an attempt to discover whether mathematical beauty is comparable to the experience derived from great art.

They’ve detailed the results in a paper titled “The experience of mathematical beauty and its neural correlates”. Here’s a bit of the abstract:

We used functional magnetic resonance imaging (fMRI) to image the activity in the brains of 15 mathematicians when they viewed mathematical formulae which they had individually rated as beautiful, indifferent or ugly. Results showed that the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain, namely field A1 of the medial orbito-frontal cortex (mOFC), as the experience of beauty derived from other sources.

BBC News puts it: “the same emotional brain centres used to appreciate art were being activated by ‘beautiful’ maths”. This is interesting, according to the authors, because it investigates the emotional response to beauty derived from “a highly intellectual and abstract source”.

As well as the open access paper, the journal website contains a sheet of the sixty mathematical formulae used in the study. Participants were asked to rate each formula on a scale of “-5 (ugly) to +5 (beautiful)”, and then two weeks later to rate each again as simply ‘ugly’, ‘neutral’ or ‘beautiful’ while in a scanner. The results of these ratings are available in an Excel data sheet.

This free access to research data means we can add to the sum total of human knowledge, namely by presenting a roundup of the most beautiful and most ugly equations!