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| The '''De Bruijn–Newman constant''', denoted by '''Λ''' and named after [[Nicolaas Govert de Bruijn]] and [[Charles M. Newman]], is a [[mathematical constant]] defined via the zeros of a certain [[function (mathematics)|function]] ''H''(''λ'', ''z''), where ''λ'' is a [[real number|real]] parameter and ''z'' is a [[complex number|complex]] variable. ''H'' has only real zeros if and only if ''λ'' ≥ Λ. The constant is closely connected with [[Riemann hypothesis|Riemann's hypothesis]] concerning the zeros of the [[Riemann zeta function|Riemann zeta-function]]. In brief, the Riemann hypothesis is equivalent to the conjecture that Λ ≤ 0. | | The writer is known as Irwin Wunder but it's not the most masucline title out there. For many years he's been operating as a receptionist. His family members life in South Dakota but his spouse desires them to move. One of the extremely very best issues in the globe for me is to do aerobics and I've been performing it for fairly a whilst.<br><br>Here is my web site; [http://bit.ly/1pABYYJ meal delivery service] |
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| [[Nicolaas Govert de Bruijn|De Bruijn]] showed in 1950 that ''H'' has only real zeros if ''λ'' ≥ 1/2, and moreover, that if ''H'' has only real zeros for some λ, ''H'' also has only real zeros if λ is replaced by any larger value. [[Charles M. Newman|Newman]] proved in 1976 the existence of a constant Λ for which the "if and only if" claim holds; and this then implies that Λ is unique. Newman conjectured that Λ ≥ 0, an intriguing counterpart to the Riemann hypothesis. Serious calculations on lower bounds for Λ have been made since 1988 and—as can be seen from the table—are still being made:
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| {| class="wikitable"
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| !Year||Lower bound on Λ
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| |1988 || −50
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| |-
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| |1991 || −5
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| |-
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| |1990 || −0.385
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| |-
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| |1994 || −4.379{{e|−6}}
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| |-
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| |1993 || −5.895{{e|−9}}
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| |-
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| |2000 || −2.7{{e|−9}}
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| |}
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| Since <math> H(\lambda , z) </math> is just the Fourier transform of <math> F(e^{\lambda x}\Phi) </math> then ''H'' has the [[Wiener–Hopf representation]]:
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| :<math> \xi (1/2+iz)= A\sqrt \pi (\lambda)^{-1} \int_{-\infty}^\infty e^{\frac{-1}{4\lambda}(x-z)^{2}} H(\lambda , x) \, dx </math>
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| which is only valid for lambda positive or 0, it can be seen that in the limit lambda tends to zero then <math> H(0,x)=\xi(1/2+ix) </math> for the case Lambda is negative then H is defined so:
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| :<math> H(z,\lambda)=B\sqrt \pi (\lambda)^{-1} \int_{-\infty}^\infty e^{\frac{-1}{4\lambda}(x-z)^{2}} \xi(1/2+ix) \, dx </math>
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| where ''A'' and ''B'' are real constants.
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| ==References==
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| *{{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 |format=pdf |accessdate=June 1, 2012 | zbl=0807.11059 }}
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| *{{cite journal |first1=N.G. |last1=de Bruijn | authorlink=Nicolaas Govert de Bruijn | title=The Roots of Triginometric Integrals |journal=Duke Math. J. |volume=17 |pages=197–226 |year=1950 | zbl=0038.23302 }}
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| *{{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 }}
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| *{{cite journal |first1=A.M. |last1=Odlyzko | authorlink=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 }}
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| ==External links==
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| * {{MathWorld|urlname=deBruijn-NewmanConstant|title=de Bruijn–Newman Constant}}
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| {{DEFAULTSORT:De Bruijn-Newman constant}}
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| [[Category:Mathematical constants]]
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| [[Category:Analytic number theory]]
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