2,924 research outputs found
P\'olya-Vinogradov and the least quadratic nonresidue
It is well-known that cancellation in short character sums (e.g. Burgess'
estimates) yields bounds on the least quadratic nonresidue. Scant progress has
been made on short character sums since Burgess' work, so it is desirable to
find a new approach to nonresidues. The goal of this note is to demonstrate a
new line of attack via long character sums, a currently active area of
research. Among other results, we demonstrate that improving the constant in
the P\'{o}lya-Vinogradov inequality would lead to significant progress on
nonresidues. Moreover, conditionally on a conjecture on long character sums, we
show that the least nonresidue for any odd primitive character (mod ) is
bounded by .Comment: 9 pages; a few small corrections from the previous versio
The distribution of the maximum of character sums
We obtain explicit bounds on the moments of character sums, refining
estimates of Montgomery and Vaughan. As an application we obtain results on the
distribution of the maximal magnitude of character sums normalized by the
square root of the modulus, finding almost double exponential decay in the tail
of this distribution.Comment: 16 pages, 1 figure, new version with correction
Ferromagnetic transition in a one-dimensional spin-orbit-coupled metal and its mapping to a critical point in smectic liquid crystals
We study the quantum phase transition between a paramagnetic and
ferromagnetic metal in the presence of Rashba spin-orbit coupling in one
dimension. Using bosonization, we analyze the transition by means of
renormalization group, controlled by an -expansion around the
upper critical dimension of two. We show that the presence of Rashba spin-orbit
coupling allows for a new nonlinear term in the bosonized action, which
generically leads to a fluctuation driven first-order transition. We further
demonstrate that the Euclidean action of this system maps onto a classical
smectic-A -- C phase transition in a magnetic field in two dimensions. We show
that the smectic transition is second-order and is controlled by a new critical
point.Comment: 16 pages, 4 figures, 1 tabl
Preventing Depression: Using Conflict-of-Interest in Evidence-Based Medicine to Teach Information Literacy
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