Dirichlet's Theorem for polynomial rings

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many polynomials c(X)\in F[X] such that a(X) + b(X)c(X) is irreducible of degree n, provided that F has a separable extension of degree n.Comment: final versio

Correction to ‘Shifted convolution and the Titchmarsh divisor problem over Fq[t]

PublishedCorrection to original article: Phil. Trans. R. Soc. A 373, 20140308 (28 April 2015; Published online 23 March 2015) (doi:10.1098/rsta.2014.0308). Two of the equations in the original article contained a typographical error. The author's accepted manuscript of the original article is available in this repository via: http://hdl.handle.net/10871/2062

Shifted convolution and the Titchmarsh divisor problem over F_q[t]

In this paper we solve a function field analogue of classical problems in analytic number theory, concerning the auto-correlations of divisor functions, in the limit of a large finite field.Comment: 22 pages, updated versio

Permanence criteria for semi-free profinite groups

We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x,y)) have free absolute Galois groups.Comment: 24 page

Pair-breaking effect on mesoscopic persistent currents

We consider the contribution of superconducting fluctuations to the mesoscopic persistent current (PC) of an ensemble of normal metallic rings, made of a superconducting material whose low bare transition temperature $T^{0}_{c}$ is much smaller than the Thouless energy $E_{c}$. The effect of pair breaking is introduced via the example of magnetic impurities. We find that over a rather broad range of pair-breaking strength $\hbar/\tau_{s}$, such that $T_c^0 \lesssim \hbar/\tau_s \lesssim E_c$, the superconducting transition temperature is normalized down to minute values or zero while the PC is hardly affected. This may provide an explanation for the magnitude of the average PC's in copper and gold, as well as a way to determine their $T^0_c$'s. The dependence of the current and the dominant superconducting fluctuations on $E_c\tau_s$ and on the ratio between $E_c$ and the temperature is analyzed. The measured PC's in copper (gold) correspond to $T^0_c$ of a few (a fraction of) mK

Effect of pair-breaking on mesoscopic persistent currents well above the superconducting transition temperature

We consider the mesoscopic normal persistent current (PC) in a very low-temperature superconductor with a bare transition temperature $T_c^0$ much smaller than the Thouless energy $E_c$. We show that in a rather broad range of pair-breaking strength, $T_c^0 \lesssim \hbar/\tau_s \lesssim E_c$, the transition temperature is renormalized to zero, but the PC is hardly affected. This may provide an explanation for the magnitude of the average PC's in the noble metals, as well as a way to determine their $T_c^0$'s.Comment: 4 pages, 2 figure