Complements to Li's criterion for the Riemann hypothesis
Complements to Li's criterion for the Riemann hypothesis
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DOI：
10.1006/jnth.1999.2392
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发表时间：
1999-08-01
影响因子：
0.7
通讯作者：
Lagarias, JC
中科院分区：
文献类型：
--
作者：
Bombieri, E;Lagarias, JC
在最近的一篇论文中，Xian-Jin Li 表明，当且仅当 lambda(n) = Sigma(rho)[1 - (1 - 1/rho)(n)] 且 lambda(n) > 0（对于 n = 1,2, 3, ...）时，黎曼假设成立，其中 rho 运行在黎曼 zeta 函数的复零点上。我们表明，Li 准则是任意复数 rho 多重集的一般不等式集的结果，因此并不特定于 zeta 函数。我们还通过Guinand-Weil显式公式给出了Li论文中数字lambda(n)的算术公式，并将lambda(n)的猜想正性与黎曼猜想的威尔准则联系起来。 (C) 1999 年学术出版社。
In a recent paper Xian-Jin Li showed that the Riemann Hypothesis holds if and only if lambda(n) = Sigma(rho)[1 - (1 - 1/rho)(n)] has lambda(n) > 0 for n = 1,2, 3, ... where rho runs over the complex zeros of the Riemann zeta function. We show that Li's criterion follows as a consequence of a general set of inequalities for an arbitrary multiset of complex numbers rho and therefore is not specific to zeta functions. We also give an arithmetic formula for the numbers lambda(n) in Li's paper, via the Guinand-Weil explicit formula, and relate the conjectural positivity of lambda(n) to Well's criterion for the Riemann Hypothesis. (C) 1999 Academic Press.