Li's criterion

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 $&xi;$ function is given by
 * $$\xi (s)=\frac{1}{2}s(s-1) \pi^{-s/2} \Gamma \left(\frac{s}{2}\right) \zeta(s)$$

where ζ is the Riemann zeta function. Consider the sequence


 * $$\lambda_n = \frac{1}{(n-1)!} \left. \frac{d^n}{ds^n}

\left[s^{n-1} \log \xi(s) \right] \right|_{s=1}.$$

Li's criterion is then the statement that


 * the Riemann hypothesis is equivalent to the statement that $$\lambda_n > 0$$ for every positive integer $$n$$.

The numbers $$\lambda_n$$ (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:


 * $$\lambda_n=\sum_{\rho} \left[1-

\left(1-\frac{1}{\rho}\right)^n\right]$$

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


 * $$\sum_\rho = \lim_{N\to\infty} \sum_{|\operatorname{Im}(\rho)|\le N}.$$

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

The positivity of $$\lambda_n$$ has been verified up to $$n = 10^5$$ by direct computation.

Proof
Note that $$\left|1-\frac{1}{\rho}\right| < 1 \Leftrightarrow |\rho-1| < |\rho| \Leftrightarrow Re(\rho) > 1/2$$.

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

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

Write the Taylor series $$\frac{\phi'(z)}{\phi(z)} = \sum_{n=0}^\infty c_n z^n $$. Since
 * $$\log \phi(z) = \sum_\rho{ \log \left(1-\frac{1}{\rho (1-z)}\right)} = \sum_\rho{ \log\left(1-\frac{1}{\rho}-z\right)-\log(1-z)}$$

we have
 * $$\frac{\phi'(z)}{\phi(z)} = \sum_\rho{ \frac{1}{1-z}-\frac{1}{1-\frac{1}{\rho}-z}}$$

so that
 * $$c_n = \sum_\rho{ 1-\left(1-\frac{1}{\rho}\right)^{-n-1}} = \sum_\rho {1-\left(1-\frac{1}{1-\rho}\right)^{n+1}}$$.

Finally, if each zero $$\rho$$ comes paired with its complex conjugate $$\bar{\rho}$$, then we may combine terms to get

The condition $$Re(\rho) \le 1/2$$ then becomes equivalent to $$\lim \sup_{n \to \infty} |c_n|^{1/n} \le 1$$. The right-hand side of ($$) is obviously nonnegative when both $$n \ge 0$$ and $$\left|1-\frac{1}{1-\rho}\right| \le 1 \Leftrightarrow\left|1-\frac{1}{\rho}\right| \ge 1 \Leftrightarrow Re(\rho) \le 1/2$$. Conversely, ordering the $$\rho$$ by $$\left|1-\frac{1}{1-\rho}\right|$$, we see that the largest $$\left|1-\frac{1}{1-\rho}\right|> 1$$ term ($$ \Leftrightarrow Re(\rho) > 1/2$$) dominates the sum as $$n \to \infty$$, and hence $$c_n$$ becomes negative sometimes.

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


 * $$\sum_\rho \frac{1+\left|\operatorname{Re}(\rho)\right|}{(1+|\rho|)^2} < \infty.$$

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


 * One has $$\operatorname{Re}(\rho) \le 1/2$$ for every &rho; if and only if
 * $$\sum_\rho\operatorname{Re}\left[1-\left(1-\frac{1}{\rho}\right)^{-n}\right]

\ge 0$$
 * 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 &minus; s. Namely, if, whenever &rho; is in R, then both the complex conjugate $$\overline{\rho}$$ and $$1-\rho$$ are in R, then Li's criterion can be stated as:


 * One has Re(&rho;) = 1/2 for every &rho; if and only if


 * $$\sum_\rho\left[1-\left(1-\frac{1}{\rho}\right)^n \right] \ge 0$$


 * for all positive integers n.

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