Chebyshev's Bias against Splitting and Principal Primes in Global Fields

A reason for the emergence of Chebyshev's bias is investigated. The Deep Riemann Hypothesis (DRH) enables us to reveal that the bias is a natural phenomenon for making a well-balanced disposition of the whole sequence of primes, in the sense that the Euler product converges at the center. By means of a weighted counting function of primes, we succeed in expressing magnitudes of the deflection by a certain asymptotic formula under the assumption of DRH, which gives a new formulation of Chebyshev's bias. For any Galois extension of global fields and for any element $\sigma$ in the Galois group, we establish a criterion of the bias of primes whose Frobenius elements are equal to $\sigma$ under the assumption of DRH. As an application we obtain a bias toward non-splitting and non-principle primes in abelian extensions under DRH. In positive characteristic cases, DRH is proved, and all these results hold unconditionally

Magnetic Reynolds number dependence of reconnection rate and flow structure of the self-similar evolution model of fast magnetic reconnection

This paper investigates Magnetic Reynolds number dependence of the ``self-similar evolution model'' (Nitta et al. 2001) of fast magnetic reconnection. I focused my attention on the flow structure inside and around the reconnection outflow, which is essential to determine the entire reconnection system (Nitta et al. 2002). The outflow is consist of several regions divided by discontinuities, e.g., shocks, and it can be treated by a shock-tube approximation (Nitta 2004). By solving the junction conditions (e.g., Rankine-Hugoniot condition), the structure of the reconnection outflow is obtained. Magnetic reconnection in most astrophysical problems is characterized by a huge dynamic range of its expansion ($sim 10^7$ for typical solar flares) in a free space which is free from any influence of external circumstances. Such evolution results in a spontaneous self-similar expansion which is controlled by two intrinsic parameters: the plasma-$beta$ and the magnetic Reynolds number. The plasma-$beta$ dependence had been investigated in our previous paper. This paper newly clarifies the relation between the reconnection rate and the inflow structure just outside the Petschek-like slow shock: As the magnetic Reynolds number increases, strongly converging inflow toward the Petschek-like slow shock forms, and it significantly reduces the reconnection rate.Comment: 16 pages. to appear in ApJ (2006 Jan. 20 issue

Continuous transition from fast magnetic reconnection to slow reconnection and change of the reconnection system structure

This paper analytically investigates a series of two-dimensional MHD reconnection solutions over a wide variation of magnetic Reynolds number ($R_{em}^*$). A new series of solutions explains a continuous transition from Petschek-like fast regime to a Sweet-Parker-like slow regime. The inflow region is obtained from a Grad-Shafranov analysis used by Nitta et al. 2002 and the outflow region from a shock-tube approximation used by Nitta 2004, 2006. A single X-point (Petschek-like) solution forms for a sufficiently small $R_{em}^*$. As $R_{em}^*$ gradually increases, the solutions shifts to an X-O-X solution with a magnetic island between two X-points. When $R_{em}^*$ increases further, the island collapses to a new elongated current sheet with Y-points at both ends (Sweet-Parker-like). These reconnection structures expand self-similarly as time proceeds. As $R_{em}^*$ increases, the reconnection rate and the reducible fraction of the initial magnetic energy of the system decrease as power-law functions of $R_{em}^*$.Comment: 19 pages, 12 figure