Communications in Number Theory and Physics

Volume 2 (2008)

Number 3

Correlations of eigenvalues and Riemann zeros

Pages: 477 – 536



John Conrey (Brian)

Nina Snaith (Claire)


Interest in comparing the statisticsof the zeros of the Riemann zeta function with random matrix theory datesback to the 1970s and the work of Montgomery and Dyson. Twelve years agoRudnick and Sarnak and, independently, Bogomolny and Keating showed thatthe $n$-point correlation function of the Riemann zeros, correctly scaledand in the limit of infinite height on the critical line, agrees with thescaling limit of the $n$-correlation of eigenvalues of random unitarymatrices. The former piece of work holds only for a restricted class oftest functions, and the latter relies on a heuristic method and theconjectures of Hardy and Littlewood. Neither tells us more than theasymptotic limit for the general $n$-correlation. In this article we usethe ratios conjecture for average values of the Riemann zeta function toproduce the lower order terms in a very precise formula for the$n$-correlation of the Riemann zeros. The same method can be appliedrigorously in the random matrix case, yielding a formula which showsidentical structure (though with none of the arithmetic details) to thecorrelations of the Riemann zeros—something which cannot be seen from theclassical determinantal formula for the random matrix correlationfunctions.

Published 1 January 2008