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

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

where ζ is the Riemann zeta function. Consider the sequence

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

Li's criterion is then the statement that

the Riemann hypothesis is equivalent to the statement that ${\displaystyle \lambda _{n}>0}$ for every positive integer ${\displaystyle n}$.

The numbers ${\displaystyle \lambda _{n}}$ (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:

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

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

${\displaystyle \sum _{\rho }=\lim _{N\to \infty }\sum _{|\operatorname {Im} (\rho )|\leq N}.}$

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

The positivity of ${\displaystyle \lambda _{n}}$ has been verified up to ${\displaystyle n=10^{5}}$ by direct computation.

Proof

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

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

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

Write the Taylor series ${\displaystyle {\frac {\phi '(z)}{\phi (z)}}=\sum _{n=0}^{\infty }c_{n}z^{n}}$. Since

${\displaystyle \log \phi (z)=\sum _{\rho }{\log \left(1-{\frac {1}{\rho (1-z)}}\right)}=\sum _{\rho }{\log \left(1-{\frac {1}{\rho }}-z\right)-\log(1-z)}}$

we have

${\displaystyle {\frac {\phi '(z)}{\phi (z)}}=\sum _{\rho }{{\frac {1}{1-z}}-{\frac {1}{1-{\frac {1}{\rho }}-z}}}}$

so that

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

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

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

(1)

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

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

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

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

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

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 ${\displaystyle {\overline {\rho }}}$ and ${\displaystyle 1-\rho }$ are in R, then Li's criterion can be stated as:

One has Re(ρ) = 1/2 for everyρif and only if

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

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.