and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function
is the super-exponentially decaying function and
is the unique real number with the property that
The constant is closely connected with Riemann hypothesis.
Indeed, the Riemann hypothesis is equivalent to the conjecture that
[1] Brad Rodgers and Terence Tao proved that
[2] A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner.
[4] Newman proved in 1976 the existence of a constant
for which the "if and only if" claim holds; and this then implies that
,[5] which was proven forty years later, by Brad Rodgers and Terence Tao in 2018.
De Bruijn's upper bound of
was not improved until 2008, when Ki, Kim and Lee proved
, making the inequality strict.
[6] In December 2018, the 15th Polymath project improved the bound to
[7][8][9] A manuscript of the Polymath work was submitted to arXiv in late April 2019,[10] and was published in the journal Research In the Mathematical Sciences in August 2019.
[11] This bound was further slightly improved in April 2020 by Platt and Trudgian to