De Bruijn–Newman constant

Template:Short description Template:DistinguishTemplate:For Template:Lowercase title The de Bruijn–Newman constant, denoted by <math>\Lambda</math> and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function <math>H(\lambda,z)</math>, where <math>\lambda</math> is a real parameter and <math>z</math> is a complex variable. More precisely,

<math>H(\lambda, z):=\int_{0}^{\infty} e^{\lambda u^{2}} \Phi(u) \cos (z u) \, du</math>,

where <math>\Phi</math> is the super-exponentially decaying function

<math>\Phi(u) = \sum_{n=1}^{\infty} (2\pi^2n^4e^{9u}-3\pi n^2 e^{5u} ) e^{-\pi n^2 e^{4u}}</math>

and <math>\Lambda</math> is the unique real number with the property that <math>H</math> has only real zeros if and only if <math>\lambda\geq \Lambda</math>.

The constant is closely connected with Riemann hypothesis. Indeed, the Riemann hypothesis is equivalent to the conjecture that <math>\Lambda\leq 0</math>.<ref name="tao2">{{#invoke:citation/CS1|citation |CitationClass=web }} (announcement post)</ref> Brad Rodgers and Terence Tao proved that <math>\Lambda\geq 0</math>, so the Riemann hypothesis is equivalent to <math>\Lambda=0</math>.<ref name=":0">Template:Cite journal</ref> A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner.<ref>Template:Cite arXiv</ref>

History

De Bruijn showed in 1950 that <math>H</math> has only real zeros if <math>\lambda\geq 1/2</math>, and moreover, that if <math>H</math> has only real zeros for some <math>\lambda</math>, <math>H</math> also has only real zeros if <math>\lambda</math> is replaced by any larger value.<ref name="de Bruijn roots">Template:Cite journal</ref> Newman proved in 1976 the existence of a constant <math>\Lambda</math> for which the "if and only if" claim holds; and this then implies that <math>\Lambda</math> is unique. Newman also conjectured that <math>\Lambda\geq 0</math>,<ref>Template:Cite journal</ref> which was proven forty years later, by Brad Rodgers and Terence Tao in 2018.

Upper bounds

De Bruijn's upper bound of <math>\Lambda\le 1/2</math> was not improved until 2008, when Ki, Kim and Lee proved <math>\Lambda< 1/2</math>, making the inequality strict.<ref name="Ki Kim Lee">Template:Citation (discussion).</ref>

In December 2018, the 15th Polymath project improved the bound to <math>\Lambda\leq 0.22</math>.<ref name="Polymath15">Template:Citation</ref><ref>Template:Citation</ref><ref>Template:Citation</ref> A manuscript of the Polymath work was submitted to arXiv in late April 2019,<ref name="Polymath1"> Template:Cite arXiv(preprint)</ref> and was published in the journal Research In the Mathematical Sciences in August 2019.<ref name="Polymath3">Template:Citation</ref>

This bound was further slightly improved in April 2020 by Platt and Trudgian to <math>\Lambda\leq 0.2</math>.<ref name="Platt+Trudgian"> Template:Cite journal(preprint)</ref>

Historical bounds

Historical lower bounds

Year	Lower bound on Λ	Authors
1987	−50<ref>Template:Cite journal</ref>	Csordas, G.; Norfolk, T. S.; Varga, R. S.
1990	−5<ref>Template:Cite journal</ref>	te Riele, H. J. J.
1991	−0.0991<ref>Template:Cite journal</ref>	Csordas, G.; Ruttan, A.; Varga, R. S.
1993	−5.895Template:E<ref>Template:Cite journal</ref>	Csordas, G.; Odlyzko, A.M.; Smith, W.; Varga, R.S.
2000	−2.7Template:E<ref>Template:Cite journal</ref>	Odlyzko, A.M.
2011	−1.1Template:E<ref>Template:Cite journal</ref>	Saouter, Yannick; Gourdon, Xavier; Demichel, Patrick
2018	≥0<ref name=":0" />	Rodgers, Brad; Tao, Terence

Historical upper bounds

Year	Upper bound on Λ	Authors
1950	≤ 1/2<ref name="de Bruijn roots" />	de Bruijn, N.G.
2008	< 1/2<ref name="Ki Kim Lee" />	Ki, H.; Kim, Y-O.; Lee, J.
2019	≤ 0.22<ref name="Polymath15" />	Polymath, D.H.J.
2020	≤ 0.2<ref name="Platt+Trudgian" />	Platt, D.; Trudgian, T.

External links

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