Fourier transforms with only real zeros
Fourier transforms with only real zeros
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DOI：
10.1090/s0002-9939-1976-0434982-5
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发表时间：
1976-02
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影响因子：
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通讯作者：
C. Newman
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文献类型：
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作者：
C. Newman
一类偶的，非负的，有限的p在实数线上使得对于任意b > exp(bt2) dp(t)的傅里叶变换只有实数0是完全确定的。然后将这个结果应用于黎曼假设。1. 主要的结果。确定傅里叶变换是否只有实零的问题出现在数学的两个截然不同的领域：数论和数学物理。在数论中，这个问题与黎曼假设密切相关[T，第10章]，而在数学物理中，它与统计力学的Lee-Yang定理和量子场论密切相关[SGj， [NI], [N3]；参见[P，第424-426页]中关于这两个主题之间历史联系的讨论。本文的结果源于对某些量子场论问题的研究，但出于教学原因，我们在黎曼假设的背景下提出它们。按照标准做法，我们定义黎曼函数为（1.1）_(z) = s(s 1)7T 's /2r(s/2) (s)./2；s = iz + 2其中t (s)是黎曼函数。是严格正偶函数的傅里叶变换，
The class of even, nonnegative, finite measures p on the real line such that for any b > 0 the Fourier transform of exp(bt2) dp(t) has only real zeros is completely determined. This result is then applied to the Riemann hypothesis. 1. Main results. The problem of determining whether a Fourier transform has only real zeros arises in two rather disparate areas of mathematics: number theory and mathematical physics. In number theory, the problem is intimately associated with the Riemann hypothesis [T, Chapter 10], while in mathematical physics it is closely connected with the Lee-Yang theorem of statistical mechanics and quantum field theory [SGj, [NI], [N3]; see Kac's remarks in [P, pp. 424-426] for a discussion of the historical connection between these two topics. The results of this paper developed out of the study of certain quantum field theoretic problems, but for pedagogical reasons, we present them in the context of the Riemann hypothesis. Following standard practice, we define the Riemann xi function as (1.1) _(z) = s(s 1)7T`s/2r(S/2) (s)./2; s = iz + 2 where t (s) is the Riemann zeta function. _ is the Fourier transform of the strictly positive, even function,