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 ξ 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 )|\leq 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 $|1-{\frac {1}{\rho }}|<1\Leftrightarrow |\rho -1|<|\rho |\Leftrightarrow Re(\rho )>1/2$ .

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

Its zeros are at ${\frac {1}{1-z}}=\rho \Leftrightarrow z=1-{\frac {1}{\rho }}$ . Hence, ${\frac {\phi '(z)}{\phi (z)}}$ is holomorphic on the unit disk $|z|<1$ if and only if $|1-{\frac {1}{\rho }}|\geq 1$ i.e. iff $Re(\rho )\leq 1/2$ .

Write the Taylor series ${\frac {\phi '(z)}{\phi (z)}}=\sum _{n=0}^{\infty }c_{n}z^{n}$ and since $\log \phi (z)=\sum _{\rho }\log(1-{\frac {1}{\rho (1-z)}})=\sum _{\rho }\log(1-{\frac {1}{\rho }}-z)-\log(1-z)$ we have ${\frac {\phi '}{\phi }}(z)=\sum _{\rho }{\frac {1}{1-z}}-{\frac {1}{1-{\frac {1}{\rho }}-z}}$ so that $c_{n}=\sum _{\rho }1-(1-{\frac {1}{\rho }})^{-n-1}=\sum _{\rho }1-(1-{\frac {1}{1-\rho }})^{n+1}$ .

The condition $Re(\rho )\leq 1/2$ then becomes equivalent to $\lim \sup _{n\to \infty }|c_{n}|^{1/n}\leq 1$ . Finally, if $\rho$ comes in pair with its complex conjugate ${\bar {\rho }}$ , then $c_{n}=\sum _{\rho }Re(1-{\Big (}1-{\frac {1}{1-\rho }}{\Big )}^{n+1})$ that is obviously $\ge 0$ for any $n\geq 0$ whenever $|1-{\frac {1}{1-\rho }}|\leq 1\Leftrightarrow |1-{\frac {1}{\rho }}|\geq 1\Leftrightarrow Re(\rho )\leq 1/2$ . Conversely, ordering the $\rho$ by $|1-{\frac {1}{1-\rho }}|$ , we see that the largest $|1-{\frac {1}{1-\rho }}|>1$ term ( $\Leftrightarrow Re(\rho )>1/2$ ) dominates the sum as $n\to \infty$ , and hence $c_{n}$ 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

\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 $\operatorname {Re} (\rho )\leq 1/2$ for every ρ if and only if

\sum _{\rho }\operatorname {Re} \left[1-\left(1-{\frac {1}{\rho }}\right)^{-n}\right]\geq 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 − s. Namely, if, whenever ρ 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(ρ) = 1/2 for every ρ if and only if

\sum _{\rho }\left[1-\left(1-{\frac {1}{\rho }}\right)^{n}\right]\geq 0

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.

Bombieri, Enrico; Lagarias, Jeffrey C. (1999). "Complements to Li's criterion for the Riemann hypothesis". Journal of Number Theory. 77 (2): 274–287. MR 1702145. doi:10.1006/jnth.1999.2392.

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.

Keiper, Jerry B (1992). "Power series expansions of Riemann's 𝜉 function". Mathematics of Computation. 58 (198): 765–773. JSTOR 2153215. doi:10.2307/2153215.

Lagarias, Jeffrey C. (2004). "Li coefficients for automorphic L-functions". Annales de l'Institut Fourier. 57 (2007): 1689–1740. arXiv:math.MG/0404394 .

Li, Xian-Jin (1997). "The positivity of a sequence of numbers and the Riemann hypothesis". Journal of Number Theory. 65 (2): 325–333. MR 1462847. doi:10.1006/jnth.1997.2137.

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