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76. CMB 2009 (vol 53 pp. 95)

Ghioca, Dragos
 Towards the Full Mordell-Lang Conjecture for Drinfeld Modules Let $\phi$ be a Drinfeld module of generic characteristic, and let X be a sufficiently generic affine subvariety of $\mathbb{G_a^g}$. We show that the intersection of X with a finite rank $\phi$-submodule of $\mathbb{G_a^g}$ is finite. Keywords:Drinfeld module, Mordell-Lang conjectureCategories:11G09, 11G10

77. CMB 2009 (vol 53 pp. 140)

Mukunda, Keshav
 Pisot Numbers from $\{ 0, 1 \}$-Polynomials A \emph{Pisot number} is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a \emph{Salem number} is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial Â- one with $\{0,1\}$-coefficients Â- and shows that they form a strictly increasing sequence with limit $(1+\sqrt{5}) / 2$. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials. Categories:11R06, 11R09, 11C08

78. CMB 2009 (vol 53 pp. 187)

Ünver, Sinan
 On the Local Unipotent Fundamental Group Scheme We prove a local, unipotent, analog of Kedlaya's theorem for the pro-p part of the fundamental group of integral affine schemes in characteristic p. Category:11G25

79. CMB 2009 (vol 53 pp. 204)

Alkan, Emre; Zaharescu, Alexandru
 Corrigendum for "Consecutive large gaps in sequences defined by multiplicative constraints" No abstract. Categories:11N25, 11B05

80. CMB 2009 (vol 52 pp. 511)

Bonciocat, Anca Iuliana; Bonciocat, Nicolae Ciprian
 The Irreducibility of Polynomials That Have One Large Coefficient and Take a Prime Value We use some classical estimates for polynomial roots to provide several irreducibility criteria for polynomials with integer coefficients that have one sufficiently large coefficient and take a prime value. Keywords:Estimates for polynomial roots, irreducible polynomialsCategories:11C08, 11R09

81. 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 productsCategories:11E25, 11F11, 11F27, 30B10

82. CMB 2009 (vol 52 pp. 583)

Konstantinou, Elisavet; Kontogeorgis, Aristides
 Computing Polynomials of the Ramanujan $t_n$ Class Invariants We compute the minimal polynomials of the Ramanujan values $t_n$, where $n\equiv 11 \mod 24$, using the Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$ and have much smaller coefficients than the Hilbert polynomials. Categories:11R29, 33E05, 11R20

83. 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 zerosCategories:11M26, 11R42

84. 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

85. 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 conjectureCategories:11G50, 11G25, 11G10

86. 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

87. 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

88. 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

89. 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

90. 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, orbisurfacesCategories:58J53, 11F72

91. 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 functionsCategories:33C10, 11M06, 65B10

92. 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

93. 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

94. 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

95. 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 problemCategories:11P32, 11N36

96. 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 gapsCategories:11N25, 11B05

97. 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 $M \rightarrow \infty$. Category:11J72

98. CMB 2008 (vol 51 pp. 3)

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

99. 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 measureCategories:11C20, 11R06

100. 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
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