de Bruijn-Newman Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let $\Xi$ be the Xi Function defined by

$$\Xi(iz)={\textstyle{1\over 2}}(z^2-{\textstyle{1\over 4}})\p... ...r 2}}z+{\textstyle{1\over 4}})\zeta(z+{\textstyle{1\over 2}}).$$ (1)

$\Xi(z/2)/8$ can be viewed as the Fourier Transform of the signal

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

for $t\in\Bbb{R}\geq 0$. Then denote the Fourier Transform of $\Phi(t)e^{\lambda t^2}$ as $H(\lambda, z)$,

$${\mathcal F}[\Phi(t)e^{\lambda t^2}]=H(\lambda, z).$$ (3)

de Bruijn (1950) proved that $H$ has only Real zeros for $\lambda\geq 1/2$. C. M. Newman (1976) proved that there exists a constant $\Lambda$ such that $H$ has only Real zeros Iff $\lambda\geq\Lambda$. The best current lower bound (Csordas et al. 1993, 1994) is $\Lambda>-5.895\times 10^{-9}$. The Riemann Hypothesis is equivalent to the conjecture that $\Lambda\leq 0$.

References

Csordas, G.; Odlyzko, A.; Smith, W.; and Varga, R. S. ``A New Lehmer Pair of Zeros and a New Lower Bound for the de Bruijn-Newman Constant.'' Elec. Trans. Numer. Analysis 1, 104-111, 1993.

Csordas, G.; Smith, W.; and Varga, R. S. ``Lehmer Pairs of Zeros, the de Bruijn-Newman Constant and the Riemann Hypothesis.'' Constr. Approx. 10, 107-129, 1994.

de Bruijn, N. G. ``The Roots of Trigonometric Integrals.'' Duke Math. J. 17, 197-226, 1950.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/dbnwm/dbnwm.html

Newman, C. M. ``Fourier Transforms with only Real Zeros.'' Proc. Amer. Math. Soc. 61, 245-251, 1976.