查看“︁德布鲁因-纽曼常数”︁的源代码

'''德布鲁因-纽曼常数'''（{{lang|en|'''De Bruijn–Newman constant'''}}）是一個以特定[[函數]]''H''(''λ'',&nbsp;''z'')的零點特性有關的[[數學常數]]，用'''Λ'''來表示。函數表示式中的''λ''為[[實數]]的參數，而''z''為[[复数 (数学)|複數]]變數。''H''有實數根[[若且唯若]]''λ''&nbsp;≥&nbsp;Λ。此常數和有關[[黎曼ζ函數]]零點的[[黎曼猜想]]密切相關，簡單來說，黎曼猜想就是Λ&nbsp;≤&nbsp;0的猜想。

{| class="wikitable"
!年份||Λ的下界
|-
|1988 || &minus;50
|-
|1991 || &minus;5			
|-
|1990 || &minus;0.385			
|-
|1994 || &minus;4.379{{e|&minus;6}}
|-
|1993 || &minus;5.895{{e|&minus;9}}
|-
|2000 || &minus;2.7{{e|&minus;9}}<ref>{{cite journal |author={{le|Andrew Odlyzko}} | title=An improved bound for the de Bruijn–Newman constant |journal=Numerical Algorithms |volume=25 |pages=293–303 |year=2000 | zbl=0967.11034 }}</ref>
|-
|2011 || &minus;1.1{{e|&minus;12}}<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}}</ref>
|}

由於<math> H(\lambda , z) </math>是<math> F(e^{\lambda x}\Phi) </math>的[[傅里叶变换]]，有以下{{link-en|維納-霍普夫表示式|Wiener–Hopf representation}}：

:<math> \xi \left(\frac{1}{2}+iz\right)= A\sqrt \pi (\lambda)^{-1}  \int_{-\infty}^\infty e^{\frac{-1}{4\lambda}(x-z)^{2}} H(\lambda , x) \, dx </math>

上式只在λ為正或0時有效，在極限中λ趨近於0，而<math> H(0,x)=\xi\left(\frac{1}{2}+ix\right) </math>。若λ為負值時H定義如下：

:<math> H(z,\lambda)=B\sqrt \pi (\lambda)^{-1}  \int_{-\infty}^\infty  e^{\frac{-1}{4\lambda}(x-z)^{2}} \xi\left(\frac{1}{2}+ix\right) \, dx </math>

其中''A''和''B''都是常數。

==參考資料==
*{{cite journal |last1=Csordas |first1=G. |last2=Odlyzko |first2=A.M. |last3=Smith |first3=W. |last4=Varga |first4=R.S. |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 |format=pdf |accessdate=June 1, 2012 |zbl=0807.11059 |archive-date=2021-08-19 |archive-url=https://web.archive.org/web/20210819165508/http://www.dtc.umn.edu/~odlyzko/doc/arch/debruijn.constant.pdf |dead-url=no }}
*{{cite journal |first1=N.G. |last1=de Bruijn  | title=The Roots of Triginometric Integrals |url=https://archive.org/details/sim_duke-mathematical-journal_1950-06_17_2/page/n118 |journal=Duke Math. J. |volume=17 |pages=197–226 |year=1950 | zbl=0038.23302 }}
*{{cite journal |first1=C.M. |last1=Newman |title=Fourier Transforms with only Real Zeros |journal=Proc. Amer. Math. Soc. |volume=61 |pages=245–251 |year=1976 | zbl=0342.42007 }}
*{{cite journal |first1=A.M. |last1=Odlyzko | title=An improved bound for the de Bruijn–Newman constant |journal=Numerical Algorithms |volume=25 |pages=293-303 |year=2000 | zbl=0967.11034 }}

==外部連結==
* {{MathWorld|urlname=deBruijn-NewmanConstant|title=de Bruijn–Newman Constant}}

{{Numtheory-stub}}

[[Category:數學常數]]
[[Category:解析数论]]