Planar magnetic structures (PMSs) are periods in the solar wind during which interplanetary magnetic field vectors are nearly parallel to a single plane. One of the specific regions where PMSs have been reported are coronal mass ejection (CME)-driven sheaths. We use here an automated method to identify PMSs in 95 CME sheath regions observed in situ by the Wind and ACE spacecraft between 1997 and 2015. The occurrence and location of the PMSs are related to various shock, sheath, and CME properties. We find that PMSs are ubiquitous in CME sheaths; 85% of the studied sheath regions had PMSs with the mean duration of 6 h. In about one-third of the cases the magnetic field vectors followed a single PMS plane that covered a significant part (at least 67%) of the sheath region. Our analysis gives strong support for two suggested PMS formation mechanisms: the amplification and alignment of solar wind discontinuities near the CME-driven shock and the draping of the magnetic field lines around the CME ejecta. For example, we found that the shock and PMS plane normals generally coincided for the events where the PMSs occurred near the shock (68% of the PMS plane normals near the shock were separated by less than 20 degrees from the shock normal), while deviations were clearly larger when PMSs occurred close to the ejecta leading edge. In addition, PMSs near the shock were generally associated with lower upstream plasma beta than the cases where PMSs occurred near the leading edge of the CME. We also demonstrate that the planar parts of the sheath contain a higher amount of strong southward magnetic field than the non-planar parts, suggesting that planar sheaths are more likely to drive magnetospheric activity.

We prove that if omega is uniformly distributed on [0, 1], then as T -> infinity, t bar right arrow zeta (i omega T + it + 1/2) converges to a nontrivial random generalized function, which in turn is identified as a product of a very well-behaved random smooth function and a random generalized function known as a complex Gaussian multiplicative chaos distribution. This demonstrates a novel rigorous connection between probabilistic number theory and the theory of multiplicative chaos-the latter is known to be connected to various branches of modern probability theory and mathematical physics. We also investigate the statistical behavior of the zeta function on the mesoscopic scale. We prove that if we let delta(T) approach zero slowly enough as T -> infinity, then t bar right arrow zeta (1/2 + i delta(T)t + i omega T) is asymptotically a product of a divergent scalar quantity suggested by Selberg's central limit theorem and a strictly Gaussian multiplicative chaos. We also prove a similar result for the characteristic polynomial of a Haar distributed random unitary matrix, where the scalar quantity is slightly different but the multiplicative chaos part is identical. This says that up to scalar multiples, the zeta function and the characteristic polynomial of a Haar distributed random unitary matrix have an identical distribution on the mesoscopic scale.