Li's criterion

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In number theory, Li's criterion is a particular statement about the positivity of a certain sequence that is equivalent to the Riemann hypothesis. The criterion is named after Xian-Jin Li, who presented it in 1997. In 1999, Enrico Bombieri and Jeffrey C. Lagarias provided a generalization, showing that Li's positivity condition applies to any collection of points that lie on the Re(s) = 1/2 axis.

Definition

The Riemann ξ function is given by

[math]\displaystyle{ \xi (s)=\frac{1}{2}s(s-1) \pi^{-s/2} \Gamma \left(\frac{s}{2}\right) \zeta(s) }[/math]

where ζ is the Riemann zeta function. Consider the sequence

[math]\displaystyle{ \lambda_n = \frac{1}{(n-1)!} \left. \frac{d^n}{ds^n} \left[s^{n-1} \log \xi(s) \right] \right|_{s=1}. }[/math]

Li's criterion is then the statement that

the Riemann hypothesis is equivalent to the statement that [math]\displaystyle{ \lambda_n \gt 0 }[/math] for every positive integer [math]\displaystyle{ n }[/math].

The numbers [math]\displaystyle{ \lambda_n }[/math] (sometimes defined with a slightly different normalization) are called Keiper-Li coefficients or Li coefficients. They may also be expressed in terms of the non-trivial zeros of the Riemann zeta function:

[math]\displaystyle{ \lambda_n=\sum_{\rho} \left[1- \left(1-\frac{1}{\rho}\right)^n\right] }[/math]

where the sum extends over ρ, the non-trivial zeros of the zeta function. This conditionally convergent sum should be understood in the sense that is usually used in number theory, namely, that

[math]\displaystyle{ \sum_\rho = \lim_{N\to\infty} \sum_{|\operatorname{Im}(\rho)|\le N}. }[/math]

(Re(s) and Im(s) denote the real and imaginary parts of s, respectively.)

The positivity of [math]\displaystyle{ \lambda_n }[/math] has been verified up to [math]\displaystyle{ n = 10^5 }[/math] by direct computation.

Proof

Note that [math]\displaystyle{ \left|1-\frac{1}{\rho}\right| \lt 1 \Leftrightarrow |\rho-1| \lt |\rho| \Leftrightarrow Re(\rho) \gt 1/2 }[/math].

Then, starting with an entire function [math]\displaystyle{ f(s) = \prod_\rho{\left(1-\frac{s}{\rho}\right)} }[/math], let [math]\displaystyle{ \phi(z) = f\left(\frac{1}{1-z}\right) }[/math].

[math]\displaystyle{ \phi }[/math] vanishes when [math]\displaystyle{ \frac{1}{1-z} = \rho \Leftrightarrow z = 1-\frac{1}{\rho} }[/math]. Hence, [math]\displaystyle{ \frac{\phi'(z)}{\phi(z)} }[/math] is holomorphic on the unit disk [math]\displaystyle{ |z| \lt 1 }[/math] iff [math]\displaystyle{ \left|1-\frac{1}{\rho}\right| \ge 1 \Leftrightarrow Re(\rho) \le 1/2 }[/math].

Write the Taylor series [math]\displaystyle{ \frac{\phi'(z)}{\phi(z)} = \sum_{n=0}^\infty c_n z^n }[/math]. Since

[math]\displaystyle{ \log \phi(z) = \sum_\rho{ \log \left(1-\frac{1}{\rho (1-z)}\right)} = \sum_\rho{ \log\left(1-\frac{1}{\rho}-z\right)-\log(1-z)} }[/math]

we have

[math]\displaystyle{ \frac{\phi'(z)}{\phi(z)} = \sum_\rho{ \frac{1}{1-z}-\frac{1}{1-\frac{1}{\rho}-z}} }[/math]

so that

[math]\displaystyle{ c_n = \sum_\rho{ 1-\left(1-\frac{1}{\rho}\right)^{-n-1}} = \sum_\rho {1-\left(1-\frac{1}{1-\rho}\right)^{n+1}} }[/math].

Finally, if each zero [math]\displaystyle{ \rho }[/math] comes paired with its complex conjugate [math]\displaystyle{ \bar{\rho} }[/math], then we may combine terms to get

[math]\displaystyle{ c_n = \sum_{\rho}{Re\left(1-\left(1-\frac{1}{1-\rho}\right)^{n+1}\right)} }[/math].

 

 

 

 

(1)

The condition [math]\displaystyle{ Re(\rho) \le 1/2 }[/math] then becomes equivalent to [math]\displaystyle{ \lim \sup_{n \to \infty} |c_n|^{1/n} \le 1 }[/math]. The right-hand side of (1) is obviously nonnegative when both [math]\displaystyle{ n \ge 0 }[/math] and [math]\displaystyle{ \left|1-\frac{1}{1-\rho}\right| \le 1 \Leftrightarrow\left|1-\frac{1}{\rho}\right| \ge 1 \Leftrightarrow Re(\rho) \le 1/2 }[/math] . Conversely, ordering the [math]\displaystyle{ \rho }[/math] by [math]\displaystyle{ \left|1-\frac{1}{1-\rho}\right| }[/math], we see that the largest [math]\displaystyle{ \left|1-\frac{1}{1-\rho}\right|\gt 1 }[/math] term ([math]\displaystyle{ \Leftrightarrow Re(\rho) \gt 1/2 }[/math]) dominates the sum as [math]\displaystyle{ n \to \infty }[/math], and hence [math]\displaystyle{ c_n }[/math] becomes negative sometimes. P. Freitas (2008). "a Li–type criterion for zero–free half-planes of Riemann's zeta function". arXiv:math.MG/0507368.

A generalization

Bombieri and Lagarias demonstrate that a similar criterion holds for any collection of complex numbers, and is thus not restricted to the Riemann hypothesis. More precisely, let R = {ρ} be any collection of complex numbers ρ, not containing ρ = 1, which satisfies

[math]\displaystyle{ \sum_\rho \frac{1+\left|\operatorname{Re}(\rho)\right|}{(1+|\rho|)^2} \lt \infty. }[/math]

Then one may make several equivalent statements about such a set. One such statement is the following:

One has [math]\displaystyle{ \operatorname{Re}(\rho) \le 1/2 }[/math] for every ρ if and only if
[math]\displaystyle{ \sum_\rho\operatorname{Re}\left[1-\left(1-\frac{1}{\rho}\right)^{-n}\right] \ge 0 }[/math]
for all positive integers n.

One may make a more interesting statement, if the set R obeys a certain functional equation under the replacement s ↦ 1 − s. Namely, if, whenever ρ is in R, then both the complex conjugate [math]\displaystyle{ \overline{\rho} }[/math] and [math]\displaystyle{ 1-\rho }[/math] are in R, then Li's criterion can be stated as:

One has Re(ρ) = 1/2 for every ρ if and only if
[math]\displaystyle{ \sum_\rho\left[1-\left(1-\frac{1}{\rho}\right)^n \right] \ge 0 }[/math]
for all positive integers n.

Bombieri and Lagarias also show that Li's criterion follows from Weil's criterion for the Riemann hypothesis.

References

  • Arias de Reyna, Juan (2011). "Asymptotics of Keiper-Li coefficients". Functiones et Approximatio Commentarii Mathematici 45 (1): 7–21. doi:10.7169/facm/1317045228. 
  • Johansson, Fredrik (2015). "Rigorous high-precision computation of the Hurwitz zeta function and its derivatives". Numerical Algorithms 69 (2): 253–270. doi:10.1007/s11075-014-9893-1. 



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