201. CMB 1998 (vol 41 pp. 187)
 Loh, W. K. A.

Exponential sums on reduced residue systems
The aim of this article is to obtain an upper bound for the exponential sums
$\sum e(f(x)/q)$, where the summation runs from $x=1$ to $x=q$ with $(x,q)=1$
and $e(\alpha)$ denotes $\exp(2\pi i\alpha)$.
We shall show that the upper bound depends only on the values of $q$ and $s$, %% where $s$ is the number of terms in the polynomial $f(x)$.
Category:11L07 

202. CMB 1998 (vol 41 pp. 125)
 Boyd, David W.

Uniform approximation to Mahler's measure in several variables
If $f(x_1,\dots,x_k)$ is a polynomial with complex coefficients, the Mahler measure
of $f$, $M(f)$ is defined to be the geometric mean of $f$ over the $k$torus
$\Bbb T^k$. We construct a sequence of approximations $M_n(f)$ which satisfy
$d2^{n}\log 2 + \log M_n(f) \le \log M(f) \le \log M_n(f)$. We use these to prove
that $M(f)$ is a continuous function of the coefficients of $f$ for polynomials
of fixed total degree $d$. Since $M_n(f)$ can be computed in a finite number
of arithmetic operations from the coefficients of $f$ this also demonstrates
an effective (but impractical) method for computing $M(f)$ to arbitrary
accuracy.
Categories:11R06, 11K16, 11Y99 

203. CMB 1998 (vol 41 pp. 86)
204. CMB 1998 (vol 41 pp. 71)
 Hurrelbrink, Jurgen; Rehmann, Ulf

Splitting patterns and trace forms
The splitting pattern of a quadratic form $q$ over
a field $k$ consists of all distinct Witt indices that occur for $q$
over extension fields of $k$. In small dimensions, the complete list
of splitting patterns of quadratic forms is known. We show that
{\it all\/} splitting patterns of quadratic forms of dimension at
most nine can be realized by trace forms.
Keywords:Quadratic forms, Witt indices, generic splitting. Category:11E04 

205. CMB 1998 (vol 41 pp. 15)
 Brown, Tom; Shiue, Peter JauShyong; Yu, X. Y.

Sequences with translates containing many primes
Garrison [3], Forman [2], and Abel and Siebert [1] showed that for all positive integers
$k$ and $N$, there exists a positive integer $\lambda$ such that $n^k+\lambda$ is
prime for at least $N$ positive integers $n$. In other words, there exists $\lambda$
such that $n^k+\lambda$ represents at least $N$ primes.
We give a quantitative version of this result. We show that there exists
$\lambda \leq x^k$ such that $n^k+\lambda$, $1\leq n\leq x$, represents at
least $(\frac 1k+o(1)) \pi(x)$ primes, as $x\rightarrow \infty$. We also give some
related results.
Category:11A48 

206. CMB 1997 (vol 40 pp. 385)
207. CMB 1997 (vol 40 pp. 498)
 Selvaraj, Chikkanna; Selvaraj, Suguna

Matrix transformations based on Dirichlet convolution
This paper is a study of summability methods that are based
on Dirichlet convolution. If $f(n)$ is a function on positive integers
and $x$ is a sequence such that $\lim_{n\to \infty} \sum_{k\le n}
{1\over k}(f\ast x)(k) =L$, then $x$ is said to be {\it $A_f$summable\/}
to $L$. The necessary and sufficient condition for the matrix $A_f$ to
preserve bounded variation of sequences is established. Also, the
matrix $A_f$ is investigated as $\ell  \ell$ and $GG$ mappings. The
strength of the $A_f$matrix is also discussed.
Categories:11A25, 40A05, 40C05, 40D05 

208. CMB 1997 (vol 40 pp. 402)
 Carpenter, Jenna P.

On the Preservation of Root Numbers and the Behavior of Weil Characters Under Reciprocity Equivalence
This paper studies how the local root numbers and the Weil additive
characters of the Witt ring of a number field behave under
reciprocity equivalence. Given a reciprocity equivalence between
two fields, at each place we define a local square class which
vanishes if and only if the local root numbers are preserved. Thus
this local square class serves as a local obstruction to the
preservation of local root numbers. We establish a set of
necessary and sufficient conditions for a selection of local square
classes (one at each place) to represent a global square class.
Then, given a reciprocity equivalence that has a finite wild set,
we use these conditions to show that the local square classes
combine to give a global square class which serves as a global
obstruction to the preservation of all root numbers. Lastly, we
use these results to study the behavior of Weil characters under
reciprocity equivalence.
Categories:11E12, 11E08 

209. CMB 1997 (vol 40 pp. 376)
210. CMB 1997 (vol 40 pp. 364)
 Narayanan, Sridhar

On the nonvanishing of a certain class of Dirichlet series
In this paper,
we consider Dirichlet series with Euler products of the form
$F(s) = \prod_{p}{\bigl(1 + {a_p\over{p^s}}\bigr)}$ in $\Re(s) > 1$,
and which are regular in $\Re(s) \geq 1$ except for a pole of
order $m$ at $s = 1$.
We establish criteria for such a Dirichlet series to be nonvanishing
on the line of convergence. We also show that our results
can be applied to yield nonvanishing results for a subclass of the
Selberg class and the SatoTate conjecture.
Categories:11Mxx, 11M41 

211. CMB 1997 (vol 40 pp. 214)
 Mollin, R. A.; Goddard, B.; Coupland, S.

Polynomials of quadratic type producing strings of primes
The primary purpose of this paper is to provide necessary and
sufficient conditions for certain quadratic polynomials of negative
discriminant (which we call EulerRabinowitsch type), to produce
consecutive prime values for an initial range of input values less than
a Minkowski bound. This not only generalizes the classical work of
Frobenius, the later developments by Hendy, and the generalizations by
others, but also concludes the line of reasoning by providing a
complete list of all such primeproducing polynomials, under the
assumption of the generalized Riemann hypothesis ($\GRH$). We demonstrate
how this primeproduction phenomenon is related to the exponent of the
class group of the underlying complex quadratic field. Numerous
examples, and a remaining conjecture, are also given.
Categories:11R11, 11R09, 11R29 

212. CMB 1997 (vol 40 pp. 72)
213. CMB 1997 (vol 40 pp. 81)
 Movahhedi, A.; Salinier, A.

Une caractÃ©risation des corps satisfaisant le thÃ©orÃ¨me de l'axe principal
Resum\'e. On caract\'erise les corps $K$ satisfaisant le th\'eor\`eme
de l'axe principal \`a l'aide de propri\'et\'es des formes
carac\t\'erisation de ces m\^emes corps due \`a Waterhouse,
on retrouve \`a partir de l\`a, de fa\c{c}on \'el\'ementaire,
un r\'esultat de Becker selon lequel un pro$2$groupe qui se
r\'ealise comme groupe de Galois absolu d'un tel corps $K$ est
engendr\'e par des involutions.
ABSTRACT. We characterize general fields $K$, satisfying the
Principal Axis Theorem, by means of properties of trace forms of
the finite extensions of $K$. From this and Waterhouse's
characterization of the same fields, we rediscover, in quite an
elementary way, a result of Becker according to which a
pro$2$group which occurs as the absolute Galois group of such
a field $K$, is generated by
Categories:11E10, 12D15 
