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76. 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 \[ \prod_{n=1}^{\infty} \frac{(1-q^n)^5}{(1-q^{5n})} =1-5\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{d} d \Big) q^n \] and \[ q\prod_{n=1}^{\infty} \frac{(1-q^{5n})^5}{(1-q^{n})} =\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{n/d} d \Big) q^n. \] We give several more identities of this type.

Keywords:Power series expansions of certain infinite products
Categories:11E25, 11F11, 11F27, 30B10

77. CMB 2009 (vol 52 pp. 237)

Ghioca, Dragos
Points of Small Height on Varieties Defined over a Function Field
We obtain a Bogomolov type of result for the affine space defined over the algebraic closure of a function field of transcendence degree $1$ over a finite field.

Keywords:heights, Bogomolov conjecture
Categories:11G50, 11G25, 11G10

78. CMB 2009 (vol 52 pp. 195)

Garaev, M. Z.; Garcia, V. C.; Konyagin, S. V.
The Waring Problem with the Ramanujan $\tau$-Function, II
Let $\tau(n)$ be the Ramanujan $\tau$-function. We prove that for any integer $N$ with $|N|\ge 2$ the diophantine equation $$\sum_{i=1}^{148000}\tau(n_i)=N$$ has a solution in positive integers $n_1, n_2,\ldots, n_{148000}$ satisfying the condition $$\max_{1\le i\le 148000}n_i\ll |N|^{2/11}e^{-c\log |N|/\log\log |N|},$$ for some absolute constant $c>0.$

Categories:11B13, 11F30

79. CMB 2009 (vol 52 pp. 186)

Broughan, Kevin A.
Extension of the Riemann $\xi$-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip
If $K$ is a number field with $n_k=[k:\mathbb{Q}]$, and $\xi_k$ the symmetrized Dedekind zeta function of the field, the inequality $$\Re\,{\frac{ \xi_k'(\sigma + {\rm i} t)}{\xi_k(\sigma + {\rm i} t)}} > \frac{ \xi_k'(\sigma)}{\xi_k(\sigma)}$$ for $t\neq 0$ is shown to be true for $\sigma\ge 1+ 8/n_k^\frac{1}{3}$ improving the result of Lagarias where the constant in the inequality was 9. In the case $k=\mathbb{Q}$ the inequality is extended to $\si\ge 1$ for all $t$ sufficiently large or small and to the region $\si\ge 1+1/(\log t -5)$ for all $t\neq 0$. This answers positively a question posed by Lagarias.

Keywords:Riemann zeta function, xi function, zeta zeros
Categories:11M26, 11R42

80. CMB 2009 (vol 52 pp. 53)

Cummins, C. J.
Cusp Forms Like $\Delta$
Let $f$ be a square-free integer and denote by $\Gamma_0(f)^+$ the normalizer of $\Gamma_0(f)$ in $\SL(2,\R)$. We find the analogues of the cusp form $\Delta$ for the groups $\Gamma_0(f)^+$.

Categories:11F03, 11F22, 30F35

81. CMB 2009 (vol 52 pp. 66)

Dryden, Emily B.; Strohmaier, Alexander
Huber's Theorem for Hyperbolic Orbisurfaces
We show that for compact orientable hyperbolic orbisurfaces, the Laplace spectrum determines the length spectrum as well as the number of singular points of a given order. The converse also holds, giving a full generalization of Huber's theorem to the setting of compact orientable hyperbolic orbisurfaces.

Keywords:Huber's theorem, length spectrum, isospectral, orbisurfaces
Categories:58J53, 11F72

82. CMB 2009 (vol 52 pp. 3)

Banks, W. D.
Carmichael Numbers with a Square Totient
Let $\varphi$ denote the Euler function. In this paper, we show that for all large $x$ there are more than $x^{0.33}$ Carmichael numbers $n\le x$ with the property that $\varphi(n)$ is a perfect square. We also obtain similar results for higher powers.

Categories:11N25, 11A25

83. CMB 2009 (vol 52 pp. 63)

Dietmann, Rainer
Small Zeros of Quadratic Forms Avoiding a Finite Number of Prescribed Hyperplanes
We prove a new upper bound for the smallest zero $\mathbf{x}$ of a quadratic form over a number field with the additional restriction that $\mathbf{x}$ does not lie in a finite number of $m$ prescribed hyperplanes. Our bound is polynomial in the height of the quadratic form, with an exponent depending only on the number of variables but not on $m$.

Categories:11D09, 11E12, 11H46, 11H55

84. CMB 2009 (vol 52 pp. 117)

Poulakis, Dimitrios
On the Rational Points of the Curve $f(X,Y)^q = h(X)g(X,Y)$
Let $q = 2,3$ and $f(X,Y)$, $g(X,Y)$, $h(X)$ be polynomials with integer coefficients. In this paper we deal with the curve $f(X,Y)^q = h(X)g(X,Y)$, and we show that under some favourable conditions it is possible to determine all of its rational points.

Categories:11G30, 14G05, 14G25

85. CMB 2008 (vol 51 pp. 497)

Borwein, Peter; Choi, Kwok-Kwong Stephen; Mercer, Idris
Expected Norms of Zero-One Polynomials
Let $\cA_n = \big\{ a_0 + a_1 z + \cdots + a_{n-1}z^{n-1} : a_j \in \{0, 1 \ } \big\}$, whose elements are called \emf{zero-one polynomials} and correspond naturally to the $2^n$ subsets of $[n] := \{ 0, 1, \ldots, n-1 \}$. We also let $\cA_{n,m} = \{ \alf(z) \in \cA_n : \alf(1) = m \}$, whose elements correspond to the ${n \choose m}$ subsets of~$[n]$ of size~$m$, and let $\cB_n = \cA_{n+1} \setminus \cA_n$, whose elements are the zero-one polynomials of degree exactly~$n$. Many researchers have studied norms of polynomials with restricted coefficients. Using $\norm{\alf}_p$ to denote the usual $L_p$ norm of~$\alf$ on the unit circle, one easily sees that $\alf(z) = a_0 + a_1 z + \cdots + a_N z^N \in \bR[z]$ satisfies $\norm{\alf}_2^2 = c_0$ and $\norm{\alf}_4^4 = c_0^2 + 2(c_1^2 + \cdots + c_N^2)$, where $c_k := \sum_{j=0}^{N-k} a_j a_{j+k}$ for $0 \le k \le N$. If $\alf(z) \in \cA_{n,m}$, say $\alf(z) = z^{\beta_1} + \cdots + z^{\beta_m}$ where $\beta_1 < \cdots < \beta_m$, then $c_k$ is the number of times $k$ appears as a difference $\beta_i - \beta_j$. The condition that $\alf \in \cA_{n,m}$ satisfies $c_k \in \{0,1\}$ for $1 \le k \le n-1$ is thus equivalent to the condition that $\{ \beta_1, \ldots, \beta_m \}$ is a \emf{Sidon set} (meaning all differences of pairs of elements are distinct). In this paper, we find the average of~$\|\alf\|_4^4$ over $\alf \in \cA_n$, $\alf \in \cB_n$, and $\alf \in \cA_{n,m}$. We further show that our expression for the average of~$\|\alf\|_4^4$ over~$\cA_{n,m}$ yields a new proof of the known result: if $m = o(n^{1/4})$ and $B(n,m)$ denotes the number of Sidon sets of size~$m$ in~$[n]$, then almost all subsets of~$[n]$ of size~$m$ are Sidon, in the sense that $\lim_{n \to \infty} B(n,m)/\binom{n}{m} = 1$.

Categories:11B83, 11C08, 30C10

86. CMB 2008 (vol 51 pp. 627)

Vidanovi\'{c}, Mirjana V.; Tri\v{c}kovi\'{c}, Slobodan B.; Stankovi\'{c}, Miomir S.
Summation of Series over Bourget Functions
In this paper we derive formulas for summation of series involving J.~Bourget's generalization of Bessel functions of integer order, as well as the analogous generalizations by H.~M.~Srivastava. These series are expressed in terms of the Riemann $\z$ function and Dirichlet functions $\eta$, $\la$, $\b$, and can be brought into closed form in certain cases, which means that the infinite series are represented by finite sums.

Keywords:Riemann zeta function, Bessel functions, Bourget functions, Dirichlet functions
Categories:33C10, 11M06, 65B10

87. CMB 2008 (vol 51 pp. 561)

Kuznetsov, Alexey
Expansion of the Riemann $\Xi$ Function in Meixner--Pollaczek Polynomials
In this article we study in detail the expansion of the Riemann $\Xi$ function in Meixner--Pollaczek polynomials. We obtain explicit formulas, recurrence relation and asymptotic expansion for the coefficients and investigate the zeros of the partial sums.

Categories:41A10, 11M26, 33C45

88. CMB 2008 (vol 51 pp. 399)

Meng, Xianmeng
Linear Equations with Small Prime and Almost Prime Solutions
Let $b_1, b_2$ be any integers such that $\gcd(b_1, b_2)=1$ and $c_1|b_1|<|b_2|\leq c_2|b_1|$, where $c_1, c_2$ are any given positive constants. Let $n$ be any integer satisfying $\{gcd(n, b_i)=1$, $i=1,2$. Let $P_k$ denote any integer with no more than $k$ prime factors, counted according to multiplicity. In this paper, for almost all $b_2$, we prove (i) a sharp lower bound for $n$ such that the equation $b_1p+b_2m=n$ is solvable in prime $p$ and almost prime $m=P_k$, $k\geq 3$ whenever both $b_i$ are positive, and (ii) a sharp upper bound for the least solutions $p, m$ of the above equation whenever $b_i$ are not of the same sign, where $p$ is a prime and $m=P_k, k\geq 3$.

Keywords:sieve method, additive problem
Categories:11P32, 11N36

89. CMB 2008 (vol 51 pp. 337)

Bennett, Michael A.
Differences between Perfect Powers
We apply the hypergeometric method of Thue and Siegel to prove that if $a$ and $b$ are positive integers, then the inequality $ 0 <| a^x - b^y | < \frac{1}{4} \, \max \{ a^{x/2}, b^{y/2} \}$ has at most a single solution in positive integers $x$ and $y$. This essentially sharpens a classic result of LeVeque.

Categories:11D61, 11D45

90. CMB 2008 (vol 51 pp. 172)

Alkan, Emre; Zaharescu, Alexandru
Consecutive Large Gaps in Sequences Defined by Multiplicative Constraints
In this paper we obtain quantitative results on the occurrence of consecutive large gaps between $B$-free numbers, and consecutive large gaps between nonzero Fourier coefficients of a class of newforms without complex multiplication.

Keywords:$B$-free numbers, consecutive gaps
Categories:11N25, 11B05

91. CMB 2008 (vol 51 pp. 3)

Alaca, Ay\c{s}e; Alaca, \c{S}aban; Williams, Kenneth S.

92. CMB 2008 (vol 51 pp. 134)

Rosales, J. C.; Garc\'{\i}a-Sánchez, P. A.
Numerical Semigroups Having a Toms Decomposition
We show that the class of system proportionally modular numerical semigroups coincides with the class of numerical semigroups having a Toms decomposition.

Categories:20M14, 11D75

93. CMB 2008 (vol 51 pp. 100)

Petkov, Vesselin
Dynamical Zeta Function for Several Strictly Convex Obstacles
The behavior of the dynamical zeta function $Z_D(s)$ related to several strictly convex disjoint obstacles is similar to that of the inverse $Q(s) = \frac{1}{\zeta(s)}$ of the Riemann zeta function $\zeta(s)$. Let $\Pi(s)$ be the series obtained from $Z_D(s)$ summing only over primitive periodic rays. In this paper we examine the analytic singularities of $Z_D(s)$ and $\Pi(s)$ close to the line $\Re s = s_2$, where $s_2$ is the abscissa of absolute convergence of the series obtained by the second iterations of the primitive periodic rays. We show that at least one of the functions $Z_D(s), \Pi(s)$ has a singularity at $s = s_2$.

Keywords:dynamical zeta function, periodic rays
Categories:11M36, 58J50

94. CMB 2008 (vol 51 pp. 57)

Dobrowolski, Edward
A Note on Integer Symmetric Matrices and Mahler's Measure
We find a lower bound on the absolute value of the discriminant of the minimal polynomial of an integral symmetric matrix and apply this result to find a lower bound on Mahler's measure of related polynomials and to disprove a conjecture of D. Estes and R. Guralnick.

Keywords:integer matrices, Lehmer's problem, Mahler's measure
Categories:11C20, 11R06

95. CMB 2008 (vol 51 pp. 32)

Choi, Stephen; Zhou, Ping
On Linear Independence of a Certain Multivariate Infinite Product
Let $q,m,M \ge 2$ be positive integers and $r_1,r_2,\dots ,r_m$ be positive rationals and consider the following $M$ multivariate infinite products \[ F_i = \prod_{j=0}^\infty ( 1+q^{-(Mj+i)}r_1+q^{-2(Mj+i)}r_2+\dots + q^{-m(Mj+i)}r_m) \] for $i=0,1,\dots ,M-1$. In this article, we study the linear independence of these infinite products. In particular, we obtain a lower bound for the dimension of the vector space $\IQ F_0+\IQ F_1 +\dots + \IQ F_{M-1} + \IQ$ over $\IQ$ and show that among these $M$ infinite products, $F_0, F_1,\dots ,F_{M-1}$, at least $\sim M/m(m+1)$ of them are irrational for fixed $m$ and $M \rightarrow \infty$.


96. CMB 2007 (vol 50 pp. 594)

Laubie, François
Ramification des groupes abéliens d'automorphismes des corps $\mathbb F_q(\!(X)\!)$
Soit $q$ une puissance d'un nombre premier $p$. Dans cette note on \'etablit la g\'en\'eralisation suivante d'un th\'eor\`eme de Wintenberger : tout sous-groupe ab\'elien ferm\'e du groupe des $\mathbb F_q$-auto\-morphismes continus du corps des s\'eries formelles $\mathbb F_q(\!(X)\!)$ muni de sa filtration de ramification est un groupe filtr\'e isomorphe au groupe de Galois d'une extension ab\'elienne d'un corps local {\`a} corps r\'esiduel $\mathbb F_q$, filtr\'e par les groupes de ramification de l'extension en num\'erotation inf\'erieure.


97. CMB 2007 (vol 50 pp. 486)

Cynk, S.; Hulek, K.
Higher-Dimensional Modular\\Calabi--Yau Manifolds
We construct several examples of higher-dimensional Calabi--Yau manifolds and prove their modularity.

Categories:14G10, 14J32, 11G40

98. CMB 2007 (vol 50 pp. 409)

Luca, Florian; Shparlinski, Igor E.
Discriminants of Complex Multiplication Fields of Elliptic Curves over Finite Fields
We show that, for most of the elliptic curves $\E$ over a prime finite field $\F_p$ of $p$ elements, the discriminant $D(\E)$ of the quadratic number field containing the endomorphism ring of $\E$ over $\F_p$ is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over $\F_p$.

Categories:11G20, 11N32, 11R11

99. CMB 2007 (vol 50 pp. 334)

Chiang-Hsieh, Hung-Jen; Yang, Yifan
Determination of Hauptmoduls and Construction of Abelian Extensions of Quadratic Number Fields
We obtain Hauptmoduls of genus zero congruence subgroups of the type $\Gamma_0^+(p):=\linebreak\Gamma_0(p)+w_p$, where $p$ is a prime and $w_p$ is the Atkin--Lehner involution. We then use the Hauptmoduls, along with modular functions on $\Gamma_1(p)$ to construct families of cyclic extensions of quadratic number fields. Further examples of cyclic extension of bi-quadratic and tri-quadratic number fields are also given.

Categories:11F03, 11G16, 11R20

100. CMB 2007 (vol 50 pp. 399)

Komornik, Vilmos; Loreti, Paola
Expansions in Complex Bases
Beginning with a seminal paper of R\'enyi, expansions in noninteger real bases have been widely studied in the last forty years. They turned out to be relevant in various domains of mathematics, such as the theory of finite automata, number theory, fractals or dynamical systems. Several results were extended by Dar\'oczy and K\'atai for expansions in complex bases. We introduce an adaptation of the so-called greedy algorithm to the complex case, and we generalize one of their main theorems.

Keywords:non-integer bases, greedy expansions, beta-expansions
Categories:11A67, 11A63, 11B85
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