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

$\xi (s)=\frac{1}{2}s(s-1){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{s}{2}\right)\zeta (s)$

where ζ is the Riemann zeta function. Consider the sequence

${\lambda }_n=\frac{1}{(n-1)!}{\frac{d^n}{d{s}^n}[s^{n-1}\log \xi (s)]|}_{s=1}.$

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:

${\lambda }_n=\sum _{\rho }[1-{(1-\frac{1}{\rho })}^n]$

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

$\sum _{\rho }=\underset{N\to \mathrm{\infty }}{\lim }\sum _{|\mathrm{Im}(\rho )|\le N}.$

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

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

$\varphi$ vanishes when $\frac{1}{1-z}=\rho \iff z=1-\frac{1}{\rho }$. Hence, $\frac{{\varphi }^{\prime }(z)}{\varphi (z)}$ is holomorphic on the unit disk $|z|<1$ iff $|1-\frac{1}{\rho }|\ge 1\iff Re(\rho )\le 1/2$.

Write the Taylor series $\frac{{\varphi }^{\prime }(z)}{\varphi (z)}=\sum _{n=0}^{\mathrm{\infty }}{c}_n{z}^n$. Since

$\log \varphi (z)=\sum _{\rho }\log (1-\frac{1}{\rho (1-z)})=\sum _{\rho }\log (1-\frac{1}{\rho }-z)-\log (1-z)$

we have

$\frac{{\varphi }^{\prime }(z)}{\varphi (z)}=\sum _{\rho }\frac{1}{1-z}-\frac{1}{1-\frac{1}{\rho }-z}$

so that

$c_n=\sum _{\rho }1-{(1-\frac{1}{\rho })}^{-n-1}=\sum _{\rho }1-{(1-\frac{1}{1-\rho })}^{n+1}$.

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

$c_n=\sum _{\rho }Re(1-{(1-\frac{1}{1-\rho })}^{n+1})$.

(1)

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

$\sum _{\rho }\frac{1+|\mathrm{Re}(\rho )|}{(1+|\rho |{)}^2}<\mathrm{\infty }.$

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

One has $\mathrm{Re}(\rho )\le 1/2$ for every ρ if and only if

$\sum _{\rho }\mathrm{Re}[1-{(1-\frac{1}{\rho })}^{-n}]\ge 0$

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 − s. Namely, if, whenever ρ is in R, then both the complex conjugate $\bar{\rho }$ 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

$\sum _{\rho }[1-{(1-\frac{1}{\rho })}^n]\ge 0$

for all positive integers n.

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

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. 

• Bombieri, Enrico; Lagarias, Jeffrey C. (1999). "Complements to Li's criterion for the Riemann hypothesis". Journal of Number Theory 77 (2): 274–287. doi:10.1006/jnth.1999.2392. http://www.math.lsa.umich.edu/~lagarias/doc/bombieri.ps. 

• 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. doi:10.2307/2153215. 

• Lagarias, Jeffrey C. (2004). "Li coefficients for automorphic L-functions". Annales de l'Institut Fourier 57 (2007): 1689–1740. doi:10.5802/aif.2311. Bibcode: 2004math......4394L. 

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