Li's criterion explained

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 function is given by

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

where ζ is the Riemann zeta function. Consider the sequence

λ _{n} = (1)/((n -1 )!) \left. (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 for every positive integer n .

The numbers

λ _{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=\sum} _{\rho} \left[ 1- \left( 1- (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\toinfty} \sum _{|\operatorname{Im} (\rho)|\le N}.

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

The positivity of

λ _{n} has been verified up to n = 10 ^{5} by direct computation.

Proof

Note that

\left| 1- (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- (s)/(\rho)} \right)} , let \phi(z) = f\left( (1)/(1- z) \right) .\phi vanishes when (1)/(1- z) = \rho \Leftrightarrow z =1- (1)/(\rho) . Hence, (\phi'(z))/(\phi(z)) is holomorphic on the unit disk |z| < 1 iff \left| 1- (1)/(\rho) \right| \ge 1 \Leftrightarrow Re(\rho) \le 1 / 2 .(\phi'(z))/(\phi(z)) = \sum ^{inf ty} _{n =0} c _{n} z ^{n} . Since log \phi(z) = \sum _{\rho{} log \left( 1- (1)/(\rho ( 1- z)) \right)} = \sum _{\rho{} log \left( 1- (1)/(\rho) - z\right) -log ( 1- z)} we have (\phi'(z))/(\phi(z)) = \sum _{\rho{} (1)/(- 1- z) (1)/(1- (1)/(- z \rho)) }so that c _{n} = \sum _{\rho{} 1- \left( 1- (1)/(\rho) \right) ^{-n-1} } = \sum_\rho . Finally, if each zero \rho comes paired with its complex conjugate \bar{\rho} , then we may combine terms to get

The condition

Re(\rho) \le 1 / 2 then becomes equivalent to \lim \sup _{n} |c ^{1 /n} _{n|} \le 1 . The right-hand side of is obviously nonnegative when both n \ge 0 and \left| 1- (1)/(1- \rho) \right| \le 1 \Leftrightarrow\left| 1- (1)/(\rho) \right| \ge 1 \Leftrightarrow Re(\rho) \le 1 / 2 . Conversely, ordering the \rho by \left| 1- (1)/(1- \rho) \right| , we see that the largest \left| 1- (1)/(1- \rho) \right|> 1 term (\Leftrightarrow Re(\rho) > 1 / 2 ) dominates the sum as n \to inf ty , and hence c _{n} becomes negative sometimes.P. Freitas . a Li–type criterion for zero–free half-planes of Riemann's zeta function . 2008 . 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 =  be any collection of complex numbers ρ, not containing ρ = 1, which satisfies

\sum _{\rho} (1+ \left|\operatorname{Re)/((\rho)\right|}{( 1+ |\rho|)) ^{2}} < inf ty.

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

One has

\operatorname{Re}(\rho) \le 1 / 2 for every ρ if and only if\sum _{\rho\operatorname{Re}\left[ 1- \left( 1- (1)/(\rho)} \right) ^{-n} \right] \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

\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- (1)/(\rho)} \right) ^{n} \right] \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 . Asymptotics of Keiper-Li coefficients . . 45 . 1 . 2011 . 7–21 . 10.7169/facm/1317045228. free .
• Bombieri, Enrico . Enrico Bombieri . Lagarias, Jeffrey C. . Jeffrey C. Lagarias . Complements to Li's criterion for the Riemann hypothesis . . 77 . 2 . 1999 . 274–287 . 1702145 . 10.1006/jnth.1999.2392. free .
• Johansson, Fredrik . Rigorous high-precision computation of the Hurwitz zeta function and its derivatives . . 69 . 2 . 2015 . 253–270 . 10.1007/s11075-014-9893-1. 1309.2877 . 10344040 .
• Keiper, Jerry B . Power series expansions of Riemann's function . . 58 . 198 . 1992 . 765–773 . 10.2307/2153215. 2153215 .
• Lagarias, Jeffrey C. . Jeffrey C. Lagarias . Li coefficients for automorphic L-functions . . 57 . 2007 . 1689–1740 . 2004 . 10.5802/aif.2311 . math.MG/0404394. 2004math......4394L . 16385403 .
• Li, Xian-Jin . The positivity of a sequence of numbers and the Riemann hypothesis . . 65 . 2 . 1997 . 325–333 . 1462847 . 10.1006/jnth.1997.2137. free .