De Bruijn–Newman constant

Not to be confused with Von Mangoldt function.

For other uses of "Λ", see Lambda.

The de Bruijn–Newman constant, denoted by ${\displaystyle \Lambda }$ and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function ${\displaystyle H(\lambda ,z)}$, where ${\displaystyle \lambda }$ is a real parameter and ${\displaystyle z}$ is a complex variable. More precisely,

${\displaystyle H(\lambda ,z):=\int _{0}^{\infty }e^{\lambda u^{2}}\Phi (u)\cos(zu)\,du}$,

where ${\displaystyle \Phi }$ is the super-exponentially decaying function

${\displaystyle \Phi (u)=\sum _{n=1}^{\infty }(2\pi ^{2}n^{4}e^{9u}-3\pi n^{2}e^{5u})e^{-\pi n^{2}e^{4u}}}$

and ${\displaystyle \Lambda }$ is the unique real number with the property that ${\displaystyle H}$ has only real zeros if and only if ${\displaystyle \lambda \geq \Lambda }$.

The constant is closely connected with Riemann hypothesis. Indeed, the Riemann hypothesis is equivalent to the conjecture that ${\displaystyle \Lambda \leq 0}$.^[1] Brad Rodgers and Terence Tao proved that ${\displaystyle \Lambda \geq 0}$, so the Riemann hypothesis is equivalent to ${\displaystyle \Lambda =0}$.^[2] A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner.^[3]

History

De Bruijn showed in 1950 that ${\displaystyle H}$ has only real zeros if ${\displaystyle \lambda \geq 1/2}$, and moreover, that if ${\displaystyle H}$ has only real zeros for some ${\displaystyle \lambda }$, ${\displaystyle H}$ also has only real zeros if ${\displaystyle \lambda }$ is replaced by any larger value.^[4] Newman proved in 1976 the existence of a constant ${\displaystyle \Lambda }$ for which the "if and only if" claim holds; and this then implies that ${\displaystyle \Lambda }$ is unique. Newman also conjectured that ${\displaystyle \Lambda \geq 0}$,^[5] which was proven forty years later, by Brad Rodgers and Terence Tao in 2018.

Upper bounds

De Bruijn's upper bound of ${\displaystyle \Lambda \leq 1/2}$ was not improved until 2008, when Ki, Kim and Lee proved ${\displaystyle \Lambda <1/2}$, making the inequality strict.^[6]

In December 2018, the 15th Polymath project improved the bound to ${\displaystyle \Lambda \leq 0.22}$.^[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 ${\displaystyle \Lambda \leq 0.2}$.^[12]

Historical bounds

Historical lower bounds

Year	Lower bound on Λ	Authors
1987	−50[13]	Csordas, G.; Norfolk, T. S.; Varga, R. S.
1990	−5[14]	te Riele, H. J. J.
1991	−0.0991[15]	Csordas, G.; Ruttan, A.; Varga, R. S.
1993	−5.895×10−9[16]	Csordas, G.; Odlyzko, A.M.; Smith, W.; Varga, R.S.
2000	−2.7×10−9[17]	Odlyzko, A.M.
2011	−1.1×10−11[18]	Saouter, Yannick; Gourdon, Xavier; Demichel, Patrick
2018	≥0[2]	Rodgers, Brad; Tao, Terence

Historical upper bounds

Year	Upper bound on Λ	Authors
1950	≤ 1/2[4]	de Bruijn, N.G.
2008	< 1/2[6]	Ki, H.; Kim, Y-O.; Lee, J.
2019	≤ 0.22[7]	Polymath, D.H.J.
2020	≤ 0.2[12]	Platt, D.; Trudgian, T.