de Bruijn–Newman constant

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

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

where $\Phi$ is the super-exponentially decaying function

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

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

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

History

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

Upper bounds

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

In December 2018, the 15th Polymath project improved the bound to $\Lambda\leq 0.22$ .{{Citation|title=Effective approximation of heat flow evolution of the Riemann $\xi$ -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}}{{citation

|title=Going below $\Lambda\leq 0.22?$

|date=4 May 2018|url=https://terrytao.wordpress.com/2018/05/04/polymath15-ninth-thread-going-below-0-22/}}{{citation

|title=Zero-free regions

|url=http://michaelnielsen.org/polymath1/index.php?title=Zero-free_regions}} A manuscript of the Polymath work was submitted to arXiv in late April 2019,

{{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) and was published in the journal Research In the Mathematical Sciences in August 2019.{{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

}}

This bound was further slightly improved in April 2020 by Platt and Trudgian to $\Lambda\leq 0.2$ .

{{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)

Historical bounds

Year	Lower bound on Λ	Authors
class="wikitable" \|+ Historical lower bounds
1987	−50{{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}}	Csordas, G.; Norfolk, T. S.; Varga, R. S.
1990	−5{{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}}	te Riele, H. J. J.
1991 \|−0.0991{{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}}	Csordas, G.; Ruttan, A.; Varga, R. S.
1993	−5.895{{e\|−9}}{{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 }}	Csordas, G.; Odlyzko, A.M.; Smith, W.; Varga, R.S.
2000	−2.7{{e\|−9}}{{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 }}	Odlyzko, A.M.
2011	−1.1{{e\|−11}}{{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 }}	Saouter, Yannick; Gourdon, Xavier; Demichel, Patrick
2018	≥0	Rodgers, Brad; Tao, Terence

Year	Upper bound on Λ	Authors
class="wikitable" \|+ Historical upper bounds
1950	≤ 1/2	de Bruijn, N.G.
2008	< 1/2	Ki, H.; Kim, Y-O.; Lee, J.
2019	≤ 0.22	Polymath, D.H.J.
2020	≤ 0.2	Platt, D.; Trudgian, T.

References

External links

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

Category:Mathematical constants

Category:Analytic number theory