TY - JOUR
T1 - On the number variance of zeta zeros and a conjecture of Berry
AU - Lugar, Meghann Moriah
AU - Milinovich, Micah B.
AU - Quesada-Herrera, Emily
PY - 2023/4
Y1 - 2023/4
N2 - Assuming the Riemann hypothesis, we prove estimates for the variance of the real and imaginary part of the logarithm of the Riemann zeta function in short intervals. We give three different formulations of these results. Assuming a conjecture of Chan for how often gaps between zeros can be close to a fixed non-zero value, we prove a conjecture of Berry (1988) for the number variance of zeta zeros in the non-universal regime. In this range, Gaussian unitary ensemble statistics do not describe the distribution of the zeros. We also calculate lower order terms in the second moment of the logarithm of the modulus of the Riemann zeta function on the critical line. Assuming Montgomery's pair correlation conjecture, this establishes a special case of a conjecture of Keating and Snaith (2000).
AB - Assuming the Riemann hypothesis, we prove estimates for the variance of the real and imaginary part of the logarithm of the Riemann zeta function in short intervals. We give three different formulations of these results. Assuming a conjecture of Chan for how often gaps between zeros can be close to a fixed non-zero value, we prove a conjecture of Berry (1988) for the number variance of zeta zeros in the non-universal regime. In this range, Gaussian unitary ensemble statistics do not describe the distribution of the zeros. We also calculate lower order terms in the second moment of the logarithm of the modulus of the Riemann zeta function on the critical line. Assuming Montgomery's pair correlation conjecture, this establishes a special case of a conjecture of Keating and Snaith (2000).
KW - Riemann zeta-function
KW - Riemann hypothesis
KW - Random matrix theory
UR - http://www.scopus.com/inward/record.url?scp=85151675757&partnerID=8YFLogxK
U2 - 10.1112/mtk.12184
DO - 10.1112/mtk.12184
M3 - Article
SN - 0025-5793
VL - 69
SP - 303
EP - 348
JO - Mathematika
JF - Mathematika
IS - 2
ER -