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

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 $\left|1-{\frac {1}{\rho }}\right|\geq 1\Leftrightarrow Re(\rho )\leq 1/2$.

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

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

The condition $Re(\rho )\leq 1/2$ then becomes equivalent to $\lim \sup _{n\to \infty }|c_{n}|^{1/n}\leq 1$. The right-hand side of (1) is obviously nonnegative when both $n\geq 0$ and $\left|1-{\frac {1}{1-\rho }}\right|\leq 1\Leftrightarrow \left|1-{\frac {1}{\rho }}\right|\geq 1\Leftrightarrow Re(\rho )\leq 1/2$ . 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. 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. 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. MR 1702145.

• Johansson, Fredrik (2015). "Rigorous high-precision computation of the Hurwitz zeta function and its derivatives". Numerical Algorithms. 69 (2): 253–270. arXiv:1309.2877. doi:10.1007/s11075-014-9893-1. S2CID 10344040.

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

• Lagarias, Jeffrey C. (2004). "Li coefficients for automorphic L-functions". Annales de l'Institut Fourier. 57 (2007): 1689–1740. arXiv:math.MG/0404394. Bibcode:2004math......4394L. doi:10.5802/aif.2311. S2CID 16385403.

• 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. MR 1462847.