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76. CMB 2009 (vol 53 pp. 204)

Alkan, Emre; Zaharescu, Alexandru

77. 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

78. 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 polynomials
Categories:11C08, 11R09

79. 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

80. 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

81. 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

82. 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

83. 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

84. 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

85. 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

86. 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

87. 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

88. 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

89. 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

90. 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

91. 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

92. 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

93. 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

94. 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

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 $M \rightarrow \infty$.

Category:11J72

96. CMB 2008 (vol 51 pp. 3)

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

97. 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

98. 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

99. 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.

Category:11S15

100. 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
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