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

ξ ( s ) = 1 2 s ( s − 1 ) π − s / 2 Γ ( s 2 ) ζ ( s ) {\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

λ n = 1 ( n − 1 ) ! d n d s n [ s n − 1 log ⁡ ξ ( s ) ] | s = 1 . {\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 λ n > 0 {\displaystyle \lambda _{n}>0} for every positive integer n {\displaystyle n} .

The numbers λ n {\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:

λ n = ∑ ρ [ 1 − ( 1 − 1 ρ ) n ] {\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

∑ ρ = lim N → ∞ ∑ | Im ⁡ ( ρ ) | ≤ N . {\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 λ n {\displaystyle \lambda _{n}} has been verified up to n = 10 5 {\displaystyle n=10^{5}} by direct computation.

Proof

Note that | 1 − 1 ρ | < 1 ⇔ | ρ − 1 | < | ρ | ⇔ R e ( ρ ) > 1 / 2 {\displaystyle \left|1-{\frac {1}{\rho }}\right|<1\Leftrightarrow |\rho -1|<|\rho |\Leftrightarrow Re(\rho )>1/2} .

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

ϕ {\displaystyle \phi } vanishes when 1 1 − z = ρ ⇔ z = 1 − 1 ρ {\displaystyle {\frac {1}{1-z}}=\rho \Leftrightarrow z=1-{\frac {1}{\rho }}} . Hence, ϕ ′ ( z ) ϕ ( z ) {\displaystyle {\frac {\phi '(z)}{\phi (z)}}} is holomorphic on the unit disk | z | < 1 {\displaystyle |z|<1} iff | 1 − 1 ρ | ≥ 1 ⇔ R e ( ρ ) ≤ 1 / 2 {\displaystyle \left|1-{\frac {1}{\rho }}\right|\geq 1\Leftrightarrow Re(\rho )\leq 1/2} .

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

log ⁡ ϕ ( z ) = ∑ ρ log ⁡ ( 1 − 1 ρ ( 1 − z ) ) = ∑ ρ log ⁡ ( 1 − 1 ρ − z ) − log ⁡ ( 1 − z ) {\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

ϕ ′ ( z ) ϕ ( z ) = ∑ ρ 1 1 − z − 1 1 − 1 ρ − z {\displaystyle {\frac {\phi '(z)}{\phi (z)}}=\sum _{\rho }{{\frac {1}{1-z}}-{\frac {1}{1-{\frac {1}{\rho }}-z}}}}

so that

c n = ∑ ρ 1 − ( 1 − 1 ρ ) − n − 1 = ∑ ρ 1 − ( 1 − 1 1 − ρ ) n + 1 {\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 ρ {\displaystyle \rho } comes paired with its complex conjugate ρ ¯ {\displaystyle {\bar {\rho }}} , then we may combine terms to get

c n = ∑ ρ R e ( 1 − ( 1 − 1 1 − ρ ) n + 1 ) {\displaystyle c_{n}=\sum _{\rho }{Re\left(1-\left(1-{\frac {1}{1-\rho }}\right)^{n+1}\right)}} . (1)

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

∑ ρ 1 + | Re ⁡ ( ρ ) | ( 1 + | ρ | ) 2 < ∞ . {\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 Re ⁡ ( ρ ) ≤ 1 / 2 {\displaystyle \operatorname {Re} (\rho )\leq 1/2} for every ρ if and only if
∑ ρ Re ⁡ [ 1 − ( 1 − 1 ρ ) − n ] ≥ 0 {\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 ρ ¯ {\displaystyle {\overline {\rho }}} and 1 − ρ {\displaystyle 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
∑ ρ [ 1 − ( 1 − 1 ρ ) n ] ≥ 0 {\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.