1. CMB Online first
 Cui, Xiaohui; Wang, Chunjie; Zhu, Kehe

Area Integral Means of Analytic Functions in the Unit Disk
For an analytic function $f$ on the unit disk $\mathbb D$ we show that
the $L^2$ integral mean of $f$ on $c\lt z\lt r$ with
respect to the weighted area measure $(1z^2)^\alpha\,dA(z)$
is a logarithmically convex function of $r$ on $(c,1)$,
where $3\le\alpha\le0$ and $c\in[0,1)$. Moreover, the range
$[3,0]$ for $\alpha$ is best possible. When
$c=0$, our arguments here also simplify the proof for several
results we obtained in earlier papers.
Keywords:logarithmic convexity, area integral mean, Bergman space, Hardy space Categories:30H10, 30H20 

2. CMB 2017 (vol 60 pp. 490)
 Fiori, Andrew

A RiemannHurwitz Theorem for the Algebraic Euler Characteristic
We prove an analogue of the RiemannHurwitz theorem for computing
Euler characteristics of pullbacks of coherent sheaves through
finite maps of smooth projective varieties in arbitrary dimensions,
subject only to the condition that the irreducible components
of the branch and ramification locus have simple normal crossings.
Keywords:RiemannHurwitz, logarithmicChern class, Euler characteristic Categories:14F05, 14C17 

3. CMB Online first
 MirandaNeto, Cleto Brasileiro

A moduletheoretic characterization of algebraic hypersurfaces
In this note we prove the following surprising characterization:
if
$X\subset {\mathbb A}^n$ is an (embedded, nonempty, proper)
algebraic variety defined over a
field $k$ of characteristic zero, then $X$ is a hypersurface
if and only if the module $T_{{\mathcal O}_{{\mathbb
A}^n}/k}(X)$ of logarithmic vector fields of
$X$ is a reflexive ${\mathcal
O}_{{\mathbb A}^n}$module. As a consequence of this result,
we derive that if $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ is a
free ${\mathcal
O}_{{\mathbb A}^n}$module, which is shown to be equivalent
to the freeness of the $t$th exterior power of $T_{{\mathcal O}_{{\mathbb
A}^n}/k}(X)$ for some (in fact, any) $t\leq n$, then necessarily
$X$ is a Saito free divisor.
Keywords:hypersurface, logarithmic vector field, logarithmic derivation, free divisor Categories:14J70, 13N15, 32S22, 13C05, 13C10, 14N20, , , , , 14C20, 32M25 

4. CMB 2016 (vol 60 pp. 184)
 Pathak, Siddhi

On a Conjecture of Livingston
In an attempt to resolve a folklore conjecture of ErdÃ¶s regarding
the nonvanishing at $s=1$ of the $L$series
attached to a periodic arithmetical function with period $q$
and values in $\{ 1, 1\} $, Livingston conjectured the $\bar{\mathbb{Q}}$
 linear independence of logarithms of certain algebraic numbers.
In this paper, we disprove Livingston's conjecture for composite
$q \geq 4$, highlighting that a new approach is required to settle
ErdÃ¶s's conjecture. We also prove that the conjecture is
true for prime $q \geq 3$, and indicate that more ingredients
will be needed to settle ErdÃ¶s's conjecture for prime $q$.
Keywords:nonvanishing of Lseries, linear independence of logarithms of algebraic numbers Categories:11J86, 11J72 

5. CMB 2005 (vol 48 pp. 473)