德布鲁因-纽曼常数(De Bruijn–Newman constant)是一个以特定函数H(λ, z)的零点特性有关的数学常数,用Λ来表示。函数表示式中的λ为实数的参数,而z为复数变数。H有实数根若且唯若λ ≥ Λ。此常数和有关黎曼ζ函数零点的黎曼猜想密切相关,简单来说,黎曼猜想就是Λ ≤ 0的猜想。
由于是的傅里叶变换,有以下维纳-霍普夫表示式:
上式只在λ为正或0时有效,在极限中λ趋近于0,而。若λ为负值时H定义如下:
其中A和B都是常数。
参考资料
- Csordas, G.; Odlyzko, A.M.; Smith, W.; Varga, R.S. A new Lehmer pair of zeros and a new lower bound for the De Bruijn–Newman constant Lambda (pdf). Electronic Transactions on Numerical Analysis. 1993, 1: 104–111 [June 1, 2012]. Zbl 0807.11059. (原始内容存档 (PDF)于2021-08-19).
- de Bruijn, N.G. The Roots of Triginometric Integrals. Duke Math. J. 1950, 17: 197–226. Zbl 0038.23302.
- Newman, C.M. Fourier Transforms with only Real Zeros. Proc. Amer. Math. Soc. 1976, 61: 245–251. Zbl 0342.42007.
- Odlyzko, A.M. An improved bound for the de Bruijn–Newman constant. Numerical Algorithms. 2000, 25: 293–303. Zbl 0967.11034.
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