Editing De Bruijn–Newman constant

{{Short description|Mathematical constant}}
{{Distinguish|Von Mangoldt function}}{{For|other uses of "Λ"|Lambda}}
{{lowercase title}}
The '''de Bruijn–Newman constant''', denoted by '''<math>\Lambda</math>''' and named after [[Nicolaas Govert de Bruijn]] and [[Charles M. Newman|Charles Michael Newman]], is a [[mathematical constant]] defined via the [[zero of a function|zeros]] of a certain [[function (mathematics)|function]] <math>H(\lambda,z)</math>, where <math>\lambda</math> is a [[real number|real]] parameter and <math>z</math> is a [[complex number|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-exponential function|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">{{cite web|url=https://terrytao.wordpress.com/2018/01/19/the-de-bruijn-newman-constant-is-non-negativ/|title=The De Bruijn-Newman constant is non-negative|date=19 January 2018|access-date=2018-01-19}} (announcement post)</ref> Brad Rodgers and [[Terence Tao]] [[mathematical proof|proved]] that <math>\Lambda\geq 0</math>, so the Riemann hypothesis is equivalent to <math>\Lambda=0</math>.<ref name=":0">{{Cite journal|last1=Rodgers|first1=Brad|last2=Tao|first2=Terence|author-link2=Terence Tao|title=The de Bruijn–Newman Constant is Non-Negative|date=2020|journal=Forum of Mathematics, Pi|language=en|volume=8|pages=e6|doi=10.1017/fmp.2020.6|issn=2050-5086|doi-access=free|arxiv=1801.05914}}</ref>  A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner.<ref>{{cite arXiv |last1=Dobner |first1=Alexander |date=2020|title=A New Proof of Newman's Conjecture and a Generalization |class=math.NT |eprint=2005.05142}}</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">{{cite journal|last1=de Bruijn|first1=N.G.|author-link=Nicolaas Govert de Bruijn|year=1950|title=The Roots of Triginometric Integrals|url=https://pure.tue.nl/ws/files/1769368/597490.pdf|journal=Duke Math. J.|volume=17|issue=3|pages=197–226|doi=10.1215/s0012-7094-50-01720-0|zbl=0038.23302}}</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>{{cite journal|last1=Newman|first1=C.M.|year=1976|title=Fourier Transforms with only Real Zeros|journal=Proc. Amer. Math. Soc.|volume=61|issue=2|pages=245–251|doi=10.1090/s0002-9939-1976-0434982-5|zbl=0342.42007|doi-access=free}}</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 (mathematics)|inequality]] strict.<ref name="Ki Kim Lee">{{citation
|title = On the de Bruijn–Newman constant
|mr=2531375
|journal = [[Advances in Mathematics]]
|volume = 222
|number = 1
|pages = 281–306
|year = 2009
|issn = 0001-8708
|doi = 10.1016/j.aim.2009.04.003
|doi-access=free
|url = http://web.yonsei.ac.kr/haseo/p23-reprint.pdf
|last1 = Ki |first1 = Haseo
|last2 = Kim |first2 = Young-One
|last3 = Lee |first3 = Jungseob
}} ([https://terrytao.wordpress.com/2018/01/24/polymath-proposal-upper-bounding-the-de-bruijn-newman-constant/ discussion]).</ref>

In December 2018, the 15th [[Polymath project]] improved the bound to <math>\Lambda\leq 0.22</math>.<ref name="Polymath15">{{Citation|title=Effective approximation of heat flow evolution of the Riemann <math>\xi</math>-function, and an upper bound for the de Bruijn-Newman constant|type=preprint|author=D.H.J. Polymath|url=https://github.com/km-git-acc/dbn_upper_bound/blob/master/Writeup/debruijn.pdf|date=20 December 2018|access-date=23 December 2018}}</ref><ref>{{citation
|title=Going below <math>\Lambda\leq 0.22?</math>
|date=4 May 2018|url=https://terrytao.wordpress.com/2018/05/04/polymath15-ninth-thread-going-below-0-22/}}</ref><ref>{{citation
|title=Zero-free regions
|url=http://michaelnielsen.org/polymath1/index.php?title=Zero-free_regions}}</ref> A manuscript of the Polymath work was submitted to arXiv in late April 2019,<ref name="Polymath1">
{{cite arXiv
| eprint=1904.12438
|title =  Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn-Newman constant
|last1=Polymath  |first1=D.H.J. 
| class=math.NT
| year = 2019}}(preprint)</ref> and was published in the journal Research In the Mathematical Sciences in August 2019.<ref name="Polymath3">{{citation
|title = Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn-Newman constant
|journal = Research in the Mathematical Sciences
|volume = 6
|issue = 3
|year = 2019
|doi = 10.1007/s40687-019-0193-1
|last = Polymath
|first = D.H.J.
|arxiv = 1904.12438
|bibcode = 2019arXiv190412438P
|s2cid = 139107960
}}</ref>

This bound was further slightly improved in April 2020 by Platt and [[Timothy Trudgian|Trudgian]] to <math>\Lambda\leq 0.2</math>.<ref name="Platt+Trudgian">
{{cite journal
| arxiv=2004.09765
|title =  The Riemann hypothesis is true up to 3·1012
|last1=Platt  |first1=Dave
| last2=Trudgian  |first2=Tim
|author2-link=Timothy Trudgian
|journal = Bulletin of the London Mathematical Society
| year = 2021|volume = 53
|issue = 3
|pages = 792–797
|doi = 10.1112/blms.12460
|s2cid = 234355998
}}(preprint)</ref>

== Historical bounds ==
{| class="wikitable"

|+ Historical lower bounds

|-
!Year !! Lower bound on Λ !! Authors

|-
|1987 ||−50<ref>{{Cite journal|last1=Csordas|first1=G.|last2=Norfolk|first2=T. S.|last3=Varga|first3=R. S.|date=1987-09-01|title=A low bound for the de Bruijn-newman constant Λ|journal=Numerische Mathematik|language=en|volume=52|issue=5|pages=483–497|doi=10.1007/BF01400887|s2cid=124008641|issn=0945-3245}}</ref> || Csordas, G.; Norfolk, T. S.; Varga, R. S. 

|-
|1990 ||−5<ref>{{Cite journal|last=te Riele|first=H. J. J.|date=1990-12-01|title=A new lower bound for the de Bruijn-Newman constant|journal=Numerische Mathematik|language=en|volume=58|issue=1|pages=661–667|doi=10.1007/BF01385647|issn=0945-3245}}</ref> || te Riele, H. J. J.

|-
|1991
|−0.0991<ref>{{Cite journal|last1=Csordas|first1=G.|last2=Ruttan|first2=A.|last3=Varga|first3=R. S.|date=1991-06-01|title=The Laguerre inequalities with applications to a problem associated with the Riemann hypothesis|journal=Numerical Algorithms|language=en|volume=1|issue=2|pages=305–329|doi=10.1007/BF02142328|bibcode=1991NuAlg...1..305C|s2cid=22606966|issn=1572-9265}}</ref> || Csordas, G.; Ruttan, A.; Varga, R. S. 

|-
|1993 ||−5.895{{e|−9}}<ref>{{cite journal |last1=Csordas | first1=G. |last2=Odlyzko | first2=A.M. | author2-link=Andrew Odlyzko |last3=Smith | first3=W. | last4=Varga | first4=R.S. | author4-link=Richard S. Varga |title=A new Lehmer pair of zeros and a new lower bound for the De Bruijn–Newman constant Lambda |journal=[[Electronic Transactions on Numerical Analysis]] |volume=1 |pages=104–111 |year=1993 |url=http://www.dtc.umn.edu/~odlyzko/doc/arch/debruijn.constant.pdf |access-date=June 1, 2012 | zbl=0807.11059 }}</ref> || Csordas, G.; Odlyzko, A.M.; Smith, W.; Varga, R.S.

|-
|2000 ||−2.7{{e|−9}}<ref>{{cite journal |first1=A.M. |last1=Odlyzko | author-link=Andrew Odlyzko | title=An improved bound for the de Bruijn–Newman constant |journal=Numerical Algorithms |volume=25 |pages=293–303 |year=2000 |issue=1 | zbl=0967.11034 |bibcode=2000NuAlg..25..293O |doi=10.1023/A:1016677511798 |s2cid=5824729 }}</ref> || Odlyzko, A.M.

|-
|2011 ||−1.1{{e|−11}}<ref>{{cite journal
| last1 = Saouter | first1 = Yannick
| last2 = Gourdon | first2 = Xavier
| last3 = Demichel | first3 = Patrick
| doi = 10.1090/S0025-5718-2011-02472-5
| issue = 276
| journal = Mathematics of Computation
| mr = 2813360
| pages = 2281–2287
| title = An improved lower bound for the de Bruijn–Newman constant
| volume = 80
| year = 2011| doi-access = free
}}</ref> || Saouter, Yannick; Gourdon, Xavier; Demichel, Patrick

|-
|2018 || ≥0<ref name=":0" /> || Rodgers, Brad; Tao, Terence

|}

{| class="wikitable"

|+ 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.

|}

==References==
{{Reflist}}

==External links==
* {{MathWorld|urlname=deBruijn-NewmanConstant|title=de Bruijn–Newman Constant}}

{{DEFAULTSORT:De Bruijn-Newman constant}}
[[Category:Mathematical constants]]
[[Category:Analytic number theory]]