In this post we assume the Riemann hypothesis and the simplicity of zeroes, thus the zeroes of {\zeta} in the critical strip take the form {\frac{1}{2} \pm i \gamma_j} for some real number ordinates {0 < \gamma_1 < \gamma_2 < \dots}. From the Riemann-von Mangoldt formula, one has the asymptotic

\displaystyle \gamma_n = (1+o(1)) \frac{2\pi}{\log n} n

as {n \rightarrow \infty}; in particular, the spacing {\gamma_{n+1} - \gamma_n} should behave like {\frac{2\pi}{\log n}} on the average. However, it can happen that some gaps are unusually small compared to other nearby gaps. For the sake of concreteness, let us define a Lehmer pair to be a pair of adjacent ordinates {\gamma_n, \gamma_{n+1}} such that

\displaystyle \frac{1}{(\gamma_{n+1} - \gamma_n)^2} \geq 1.3 \sum_{m \neq n,n+1} \frac{1}{(\gamma_m - \gamma_n)^2} + \frac{1}{(\gamma_m - \gamma_{n+1})^2}. \ \ \ \ \ (1)

The specific value of constant {1.3} is not particularly important here; anything larger than {\frac{5}{4}} would suffice. An example of such a pair would be the classical pair

\displaystyle \gamma_{6709} = 7005.062866\dots

\displaystyle \gamma_{6710} = 7005.100564\dots

discovered by Lehmer. It follows easily from the main results of Csordas, Smith, and Varga that if an infinite number of Lehmer pairs (in the above sense) existed, then the de Bruijn-Newman constant {\Lambda} is non-negative. This implication is now redundant in view of the unconditional results of this recent paper of Rodgers and myself; however, the question of whether an infinite number of Lehmer pairs exist remain open.

In this post, I sketch an argument that Brad and I came up with (as initially suggested by Odlyzko) the GUE hypothesis implies the existence of infinitely many Lehmer pairs. We argue probabilistically: pick a sufficiently large number {T}, pick {n} at random from {T \log T} to {2 T \log T} (so that the average gap size is close to {\frac{2\pi}{\log T}}), and prove that the Lehmer pair condition (1) occurs with positive probability.

Introduce the renormalised ordinates {x_n := \frac{\log T}{2\pi} \gamma_n} for {T \log T \leq n \leq 2 T \log T}, and let {\varepsilon > 0} be a small absolute constant (independent of {T}). It will then suffice to show that

\displaystyle \frac{1}{(x_{n+1} - x_n)^2} \geq

\displaystyle 1.3 \sum_{m \in [T \log T, 2T \log T]: m \neq n,n+1} \frac{1}{(x_m - x_n)^2} + \frac{1}{(x_m - x_{n+1})^2}

\displaystyle + \frac{1}{6\varepsilon^2}

(say) with probability {\gg \varepsilon^4 - o(1)}, since the contribution of those {m} outside of {[T \log T, 2T \log T]} can be absorbed by the {\frac{1}{\varepsilon^2}} factor with probability {o(1)}.

As one consequence of the GUE hypothesis, we have {x_{n+1} - x_n \leq \varepsilon^2} with probability {O(\varepsilon^6)}. Thus, if {E := \{ m \in [T \log T, 2T \log T]: x_{m+1} - x_m \leq \varepsilon^2 \}}, then {E} has density {O( \varepsilon^6 )}. Applying the Hardy-Littlewood maximal inequality, we see that with probability {O(\varepsilon^6)}, we have

\displaystyle \sup_{h \geq 1} | \# E \cap [n+h, n-h] | \leq \frac{1}{10}

which implies in particular that

\displaystyle |x_m - x_n|, |x_{m} - x_{n+1}| \gg \varepsilon^2 |m-n|

for all {m \in [T \log T, 2 T \log T] \backslash \{ n, n+1\}}. This implies in particular that

\displaystyle \sum_{m \in [T \log T, 2T \log T]: |m-n| \geq \varepsilon^{-3}} \frac{1}{(x_m - x_n)^2} + \frac{1}{(x_m - x_{n+1})^2} \ll \varepsilon^{-1}

and so it will suffice to show that

\displaystyle \frac{1}{(x_{n+1} - x_n)^2}

\displaystyle \geq 1.3 \sum_{m \in [T \log T, 2T \log T]: m \neq n,n+1; |m-n| < \varepsilon^{-3}} \frac{1}{(x_m - x_n)^2} + \frac{1}{(x_m - x_{n+1})^2} + \frac{1}{5\varepsilon^2}

(say) with probability {\gg \varepsilon^4 - o(1)}.

By the GUE hypothesis (and the fact that {\varepsilon} is independent of {T}), it suffices to show that a Dyson sine process {(x_n)_{n \in {\bf Z}}}, normalised so that {x_0} is the first positive point in the process, obeys the inequality

\displaystyle \frac{1}{(x_{1} - x_0)^2} \geq 1.3 \sum_{|m| < \varepsilon^{-3}: m \neq 0,1} \frac{1}{(x_m - x_0)^2} + \frac{1}{(x_m - x_1)^2} \ \ \ \ \ (2)

with probability {\gg \varepsilon^4}. However, if we let {A > 0} be a moderately large constant (and assume {\varepsilon} small depending on {A}), one can show using {k}-point correlation functions for the Dyson sine process (and the fact that the Dyson kernel {K(x,y) = \sin(\pi(x-y))/\pi(x-y)} equals {1} to second order at the origin) that

\displaystyle {\bf E} N_{[-\varepsilon,0]} N_{[0,\varepsilon]} \gg \varepsilon^4

\displaystyle {\bf E} N_{[-\varepsilon,0]} \binom{N_{[0,\varepsilon]}}{2} \ll \varepsilon^7

\displaystyle {\bf E} \binom{N_{[-\varepsilon,0]}}{2} N_{[0,\varepsilon]} \ll \varepsilon^7

\displaystyle {\bf E} N_{[-\varepsilon,0]} N_{[0,\varepsilon]} N_{[\varepsilon,A^{-1}]} \ll A^{-3} \varepsilon^4

\displaystyle {\bf E} N_{[-\varepsilon,0]} N_{[0,\varepsilon]} N_{[-A^{-1}, -\varepsilon]} \ll A^{-3} \varepsilon^4

\displaystyle {\bf E} N_{[-\varepsilon,0]} N_{[0,\varepsilon]} N_{[-k, k]}^2 \ll k^2 \varepsilon^4 \ \ \ \ \ (3)

for any natural number {k}, where {N_{I}} denotes the number of elements of the process in {I}. For instance, the expression {{\bf E} N_{[-\varepsilon,0]} \binom{N_{[0,\varepsilon]}}{2} } can be written in terms of the three-point correlation function {\rho_3(x_1,x_2,x_3) = \mathrm{det}(K(x_i,x_j))_{1 \leq i,j \leq 3}} as

\displaystyle \int_{-\varepsilon \leq x_1 \leq 0 \leq x_2 \leq x_3 \leq \varepsilon} \rho_3( x_1, x_2, x_3 )\ dx_1 dx_2 dx_3

which can easily be estimated to be {O(\varepsilon^7)} (since {\rho_3 = O(\varepsilon^4)} in this region), and similarly for the other estimates claimed above.

Since for natural numbers {a,b}, the quantity {ab - 2 a \binom{b}{2} - 2 b \binom{a}{2} = ab (5-2a-2b)} is only positive when {a=b=1}, we see from the first three estimates that the event {E} that {N_{[-\varepsilon,0]} = N_{[0,\varepsilon]} = 1} occurs with probability {\gg \varepsilon^4}. In particular, by Markov’s inequality we have the conditional probabilities

\displaystyle {\bf P} ( N_{[\varepsilon,A^{-1}]} \geq 1 | E ) \ll A^{-3}

\displaystyle {\bf P} ( N_{[-A^{-1}, -\varepsilon]} \geq 1 | E ) \ll A^{-3}

\displaystyle {\bf P} ( N_{[-k, k]} \geq A k^{5/3} | E ) \ll A^{-4} k^{-4/3}

and thus, if {A} is large enough, and {\varepsilon} small enough, it will be true with probability {\gg \varepsilon^4} that

\displaystyle N_{[-\varepsilon,0]}, N_{[0,\varepsilon]} = 1

and

\displaystyle N_{[A^{-1}, \varepsilon]} = N_{[\varepsilon, A^{-1}]} = 0

and simultaneously that

\displaystyle N_{[-k,k]} \leq A k^{5/3}

for all natural numbers {k}. This implies in particular that

\displaystyle x_1 - x_0 \leq 2\varepsilon

and

\displaystyle |x_m - x_0|, |x_m - x_1| \gg_A |m|^{3/5}

for all {m \neq 0,1}, which gives (2) for {\varepsilon} small enough.

Remark 1 The above argument needed the GUE hypothesis for correlations up to fourth order (in order to establish (3)). It might be possible to reduce the number of correlations needed, but I do not see how to obtain the claim just using pair correlations only.