1. 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 

2. CMB 2016 (vol 60 pp. 63)
 Chang, Gyu Whan

Power Series Rings Over PrÃ¼fer $v$multiplication Domains, II
Let $D$ be an integral domain, $X^1(D)$ be the set of heightone
prime ideals of $D$,
$\{X_{\beta}\}$ and $\{X_{\alpha}\}$ be
two disjoint nonempty sets of indeterminates over $D$,
$D[\{X_{\beta}\}]$ be the polynomial ring over $D$, and
$D[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1$ be the first type
power series ring over $D[\{X_{\beta}\}]$.
Assume that $D$ is a PrÃ¼fer $v$multiplication domain (P$v$MD)
in which each proper integral $t$ideal has only finitely many
minimal prime ideals
(e.g., $t$SFT P$v$MDs, valuation domains, rings of Krull type).
Among other things, we show that if
$X^1(D) = \emptyset$ or $D_P$ is a DVR for all $P \in X^1(D)$,
then
${D[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1}_{D  \{0\}}$ is a
Krull domain.
We also prove that if $D$ is a $t$SFT P$v$MD, then the complete
integral closure of $D$ is a Krull domain and
ht$(M[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1)$ = $1$ for every
heightone maximal $t$ideal $M$ of $D$.
Keywords:Krull domain, P$v$MD, multiplicatively closed set of ideals, power series ring Categories:13A15, 13F05, 13F25 

3. CMB 2016 (vol 59 pp. 794)
 Hashemi, Ebrahim; Amirjan, R.

Zerodivisor Graphs of Ore Extensions over Reversible Rings
Let $R$ be an associative ring with identity.
First we prove some results about zerodivisor graphs of reversible
rings. Then we study the zerodivisors of the skew power series
ring $R[[x;\alpha]]$, whenever $R$ is reversible and $\alpha$compatible. Moreover, we compare the diameter and girth of the zerodivisor
graphs of $\Gamma(R)$, $\Gamma(R[x;\alpha,\delta])$ and $\Gamma(R[[x;\alpha]])$,
when
$R$ is reversible and $(\alpha,\delta)$compatible.
Keywords:zerodivisor graphs, reversible rings, McCoy rings, polynomial rings, power series rings Categories:13B25, 05C12, 16S36 

4. CMB 2015 (vol 58 pp. 704)
 Benamar, H.; Chandoul, A.; Mkaouar, M.

On the Continued Fraction Expansion of Fixed Period in Finite Fields
The Chowla conjecture
states that,
if $t$ is any given
positive integer, there are infinitely many prime positive
integers $N$ such that $\operatorname{Per} (\sqrt{N})=t$, where
$\operatorname{Per} (\sqrt{N})$
is the period length of the continued fraction expansion for
$\sqrt{N}$.
C. Friesen proved
that, for any $k\in \mathbb{N}$, there are infinitely many
squarefree integers $N$, where the continued fraction expansion
of $\sqrt{N}$ has a fixed period. In this paper, we describe all
polynomials $Q\in \mathbb{F}_q[X] $ for which the continued fraction
expansion of $\sqrt {Q}$ has a fixed period, also we give a
lower
bound of the number of monic, nonsquares polynomials $Q$ such
that $\deg Q= 2d$ and $ Per \sqrt {Q}=t$.
Keywords:continued fractions, polynomials, formal power series Categories:11A55, 13J05 

5. CMB 2015 (vol 59 pp. 119)
6. CMB 2015 (vol 59 pp. 136)
 Kajihara, Yasushi

Transformation Formulas for Bilinear Sums of Basic Hypergeometric Series
A master formula of transformation formulas for bilinear sums
of basic hypergeometric series
is proposed.
It is obtained from the author's previous results on
a transformation formula for Milne's multivariate generalization
of basic hypergeometric
series of type $A$ with different dimensions and it can be considered
as a
generalization of the WhippleSears transformation formula for
terminating balanced ${}_4 \phi_3$
series.
As an application of the master formula, the one variable cases
of some transformation formulas
for bilinear sums of basic hypergeometric series are given as
examples.
The bilinear transformation formulas seem to be new in the literature,
even in one variable case.
Keywords:bilinear sums, basic hypergeometric series Category:33D20 

7. CMB 2015 (vol 58 pp. 548)
 Lü, Guangshi; Sankaranarayanan, Ayyadurai

Higher Moments of Fourier Coefficients of Cusp Forms
Let $S_{k}(\Gamma)$ be the space of holomorphic cusp
forms of even integral weight $k$ for the full modular group
$SL(2, \mathbb{Z})$. Let
$\lambda_f(n)$, $\lambda_g(n)$, $\lambda_h(n)$ be the $n$th normalized
Fourier
coefficients of three distinct holomorphic primitive cusp forms
$f(z) \in S_{k_1}(\Gamma), g(z) \in S_{k_2}(\Gamma), h(z) \in
S_{k_3}(\Gamma)$ respectively.
In this paper we study the cancellations of sums related to arithmetic
functions, such as $\lambda_f(n)^4\lambda_g(n)^2$, $\lambda_g(n)^6$,
$\lambda_g(n)^2\lambda_h(n)^4$, and $\lambda_g(n^3)^2$ twisted
by
the arithmetic function $\lambda_f(n)$.
Keywords:Fourier coefficients of automorphic forms, Dirichlet series, triple product $L$function, Perron's formula Categories:11F30, 11F66 

8. CMB 2014 (vol 58 pp. 188)
9. CMB 2013 (vol 56 pp. 827)
 Petridis, Yiannis N.; Raulf, Nicole; Risager, Morten S.

Erratum to ``Quantum Limits of Eisenstein Series and Scattering States''
This paper provides an erratum to Y. N. Petridis,
N. Raulf, and M. S. Risager, ``Quantum Limits
of Eisenstein Series and Scattering States.'' Canad. Math. Bull., published
online 20120203, http://dx.doi.org/10.4153/CMB20112002.
Keywords:quantum limits, Eisenstein series, scattering poles Categories:11F72, 8G25, 35P25 

10. CMB 2013 (vol 56 pp. 673)
 Ayadi, K.; Hbaib, M.; Mahjoub, F.

Diophantine Approximation for Certain Algebraic Formal Power Series in Positive Characteristic
In this paper, we study rational approximations for certain algebraic power series over a finite field.
We obtain results for irrational elements of strictly positive degree
satisfying an equation of the type
\begin{equation}
\alpha=\displaystyle\frac{A\alpha^{q}+B}{C\alpha^{q}}
\end{equation}
where $(A, B, C)\in
(\mathbb{F}_{q}[X])^{2}\times\mathbb{F}_{q}^{\star}[X]$.
In particular,
we will give, under some conditions on the polynomials $A$, $B$
and $C$, well approximated elements satisfying this equation.
Keywords:diophantine approximation, formal power series, continued fraction Categories:11J61, 11J70 

11. CMB 2012 (vol 56 pp. 814)
12. CMB 2011 (vol 56 pp. 258)
 Chandoul, A.; Jellali, M.; Mkaouar, M.

The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field
Dufresnoy and Pisot characterized the smallest
Pisot number of degree $n \geq 3$ by giving explicitly its minimal
polynomial. In this paper, we translate Dufresnoy and Pisot's
result to the Laurent series case.
The
aim of this paper is to prove that the minimal polynomial
of the smallest Pisot element (SPE) of degree $n$ in the field of
formal power series over a finite field
is given by $P(Y)=Y^{n}\alpha XY^{n1}\alpha^n,$ where $\alpha$
is the least element of the finite field $\mathbb{F}_{q}\backslash\{0\}$
(as a finite total ordered set). We prove that the sequence of
SPEs of degree $n$ is decreasing and converges to $\alpha X.$
Finally, we show how to obtain explicit continued fraction
expansion of the smallest Pisot element over a finite field.
Keywords:Pisot element, continued fraction, Laurent series, finite fields Categories:11A55, 11D45, 11D72, 11J61, 11J66 

13. CMB 2011 (vol 55 pp. 571)
 Miller, A. R.; Paris, R. B.

A Generalised KummerType Transformation for the ${}_pF_p(x)$ Hypergeometric Function
In a recent paper, Miller derived a Kummertype
transformation for the generalised hypergeometric function ${}_pF_p(x)$ when pairs of
parameters differ by unity, by means of a reduction
formula for a certain KampÃ© de FÃ©riet function. An alternative and simpler derivation of this
transformation is obtained here by application of the wellknown Kummer transformation for the
confluent hypergeometric function corresponding to $p=1$.
Keywords:generalised hypergeometric series, Kummer transformation Categories:33C15, 33C20 

14. CMB 2011 (vol 55 pp. 26)
 Bertin, Marie José

A Mahler Measure of a $K3$ Surface Expressed as a Dirichlet $L$Series
We present another example of a $3$variable polynomial defining a $K3$hypersurface
and having a logarithmic Mahler measure expressed in terms of a Dirichlet
$L$series.
Keywords:modular Mahler measure, EisensteinKronecker series, $L$series of $K3$surfaces, $l$adic representations, LivnÃ© criterion, RankinCohen brackets Categories:11, 14D, 14J 

15. CMB 2011 (vol 55 pp. 60)
 Coons, Michael

Extension of Some Theorems of W. Schwarz
In this paper, we prove that a nonzero power series $F(z)\in\mathbb{C}
[\mkern3mu[ z]\mkern3mu]
$
satisfying $$F(z^d)=F(z)+\frac{A(z)}{B(z)},$$ where $d\geq 2$, $A(z),B(z)\in\mathbb{C}[z]$
with $A(z)\neq 0$ and $\deg A(z),\deg B(z)
Keywords:functional equations, transcendence, power series Categories:11B37, 11J81 

16. CMB 2009 (vol 52 pp. 627)
 Yu, Dan Sheng; Zhou, Ping; Zhou, Song Ping

On $L^{1}$Convergence of Fourier Series under the MVBV Condition
Let $f\in L_{2\pi }$ be a realvalued even function with its Fourier series $%
\frac{a_{0}}{2}+\sum_{n=1}^{\infty }a_{n}\cos nx,$ and let
$S_{n}(f,x) ,\;n\geq 1,$ be the $n$th partial sum of the Fourier series. It
is well known that if the nonnegative sequence $\{a_{n}\}$ is decreasing and
$\lim_{n\rightarrow \infty }a_{n}=0$, then%
\begin{equation*}
\lim_{n\rightarrow \infty }\Vert fS_{n}(f)\Vert _{L}=0
\text{ if
and only if }\lim_{n\rightarrow \infty }a_{n}\log n=0.
\end{equation*}%
We weaken the monotone condition in this classical result to the socalled
mean value bounded variation (MVBV) condition. The generalization of the
above classical result in realvalued function space is presented as a
special case of the main result in this paper, which gives the $L^{1}$%
convergence of a function $f\in L_{2\pi }$ in complex space. We also give
results on $L^{1}$approximation of a function $f\in L_{2\pi }$ under the
MVBV condition.
Keywords:complex trigonometric series, $L^{1}$ convergence, monotonicity, mean value bounded variation Categories:42A25, 41A50 

17. CMB 2009 (vol 52 pp. 481)
 Alaca, Ay\c{s}e; Alaca, \c{S}aban; Williams, Kenneth S.

Some Infinite Products of Ramanujan Type
In his ``lost'' notebook, Ramanujan stated two results, which are equivalent to the identities
\[
\prod_{n=1}^{\infty} \frac{(1q^n)^5}{(1q^{5n})}
=15\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{d} d \Big) q^n
\]
and
\[
q\prod_{n=1}^{\infty} \frac{(1q^{5n})^5}{(1q^{n})}
=\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{n/d} d \Big) q^n.
\]
We give several more identities of this type.
Keywords:Power series expansions of certain infinite products Categories:11E25, 11F11, 11F27, 30B10 

18. CMB 2008 (vol 51 pp. 3)
19. CMB 2007 (vol 50 pp. 284)
 McIntosh, Richard J.

Second Order Mock Theta Functions
In his last letter to Hardy, Ramanujan
defined 17 functions $F(q)$, where $q<1$. He called them mock theta
functions, because as $q$ radially approaches any point $e^{2\pi ir}$
($r$ rational), there is a theta function $F_r(q)$ with $F(q)F_r(q)=O(1)$.
In this paper we establish the relationship between two families of mock
theta functions.
Keywords:$q$series, mock theta function, Mordell integral Categories:11B65, 33D15 

20. CMB 2007 (vol 50 pp. 11)
 Borwein, David; Borwein, Jonathan

van der Pol Expansions of LSeries
We provide concise series representations for various
Lseries integrals. Different techniques are needed below and above
the abscissa of absolute convergence of the underlying Lseries.
Keywords:Dirichlet series integrals, Hurwitz zeta functions, Plancherel theorems, Lseries Categories:11M35, 11M41, 30B50 

21. CMB 2006 (vol 49 pp. 256)
 Neelon, Tejinder

A BernsteinWalsh Type Inequality and Applications
A BernsteinWalsh type inequality for $C^{\infty }$ functions of several
variables is derived, which then is applied to obtain analogs and
generalizations of the following classical theorems: (1) BochnakSiciak
theorem: a $C^{\infty }$\ function on $\mathbb{R}^{n}$ that is real
analytic on every line is real analytic; (2) ZornLelong theorem: if a
double power series $F(x,y)$\ converges on a set of lines of positive
capacity then $F(x,y)$\ is convergent; (3) AbhyankarMohSathaye theorem:
the transfinite diameter of the convergence set of a divergent series is
zero.
Keywords:BernsteinWalsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power series Categories:32A05, 26E05 

22. CMB 2005 (vol 48 pp. 147)
 Väänänen, Keijo; Zudilin, Wadim

BakerType Estimates for Linear Forms in the Values of $q$Series
We obtain lower estimates for the absolute values
of linear forms of the values of generalized Heine
series at nonzero points of an imaginary quadratic field~$\II$,
in particular of the values of $q$exponential function.
These estimates depend on the individual coefficients,
not only on the maximum of their absolute values.
The proof uses a variant of classical Siegel's method
applied to a system of functional Poincar\'etype equations
and the connection between the solutions of these functional
equations and the generalized Heine series.
Keywords:measure of linear independence, $q$series Categories:11J82, 33D15 

23. CMB 1999 (vol 42 pp. 198)
24. CMB 1998 (vol 41 pp. 86)