Multiple Dirichlet Series Research Articles

The analytic continuation and the order estimate of multiple Dirichlet series

Dans cet article, nous considerons que certaines series de Dirichlet multiples, dont nous montrons le prolongement analytique en utilisant la formule integrale de Mellin-Barnes. Des majorations de ces series sont egalement obtenues.

On the minimum modulus of a multiple Dirichlet series with monotone coefficients

We establish conditions under which the relation M(x, F) ∼ Μ(x, F) ∼ m(x, F) holds except for a small set, as ¦x¦→ +∞ for an entire function F(z) of several complex variables z ∃ ℂ (p≥2) represented by a Dirichlet series, where M(x, F) = sup{¦F(x+iy¦: y ∃ ℝp}, m(x, F) = inf{¦F(x+iy)¦: y ∃ ℝp} Μ(x, F) being the maximal term of the Dirichlet series, and x ∃ ℝp.

On the order of an entire function of several omplex variables represented by multiple dirichlet series

On the order of an entire function of several omplex variables represented by multiple dirichlet series

On the means of an entire function of several complex variables represented by multiple Dirichlet series

On the means of an entire function of several complex variables represented by multiple Dirichlet series

Behavior of maximum term of a multiple Dirichlet series defining an entire function

Behavior of maximum term of a multiple Dirichlet series defining an entire function

Some Multiple Power Series with Zero-One Coefficients

The paper is concerned with sums of the type \[S_{n,j} = \sum {x_1^{a_1 } x_2^{a_2 } \cdots x_n^{a_n } } \quad (n > 1),\] where the summation is over either \[( * )\qquad ja_i \leqq a_1 + a_2 + \cdots + a_n \quad (1 \leqq j \leqq n;1 \leqq i \leqq n)\] or \[( * * )\qquad a_1 + a_2 + \cdots + a_n = ja_i + (n - j)b_i \quad (0 \leqq j \leqq n;1 \leqq i \leqq n);\] the $a_i $ and $b_i $ are nonnegative integers. It is proved, for example, that for the first type with $j = 2$, the sum is a rational function with denominator equal to $\prod _{1 \leqq i < k \leqq n} (1 - x_i x_k )$. Several combinatorial applications are obtained by specializing the $x_i $. For example it is proved that the number of nonnegative solutions of the system \[a_1 + a_2 + \cdots + a_n = N,\quad (n - 1)a_i \leqq N,\quad (1 \leqq i \leqq n)\] is equal to the binomial coefficient \[\left( {\begin{array}{*{20}c} {k + n - s - 1} \\ {n - 1} \\ \end{array} } \right)\quad (N = k(n - 1) + s,0 \leqq s < n - 1).\] The final section of the paper is concerned with multiple Dirichlet series \[{\bf \Phi} _{n,j} = \sum {m_1^{ - s_1 } m_2^{ - s_2 } \cdots m_n^{ - s_n } } ,\] where the smmation is over all positive integers $m_i $ such that \[m_i^j \mid m_1 m_2 \cdots m_n \quad (1 \leqq j \leqq n;1 \leqq i \leqq n).\] The ${\bf \Phi} _{n,j} $ are expressed as products involving series satisfying (*); in particular \[{\bf \Phi} _{n,n - 1} = \frac{{\zeta (\alpha )\zeta (\sigma - s_1 ) \cdots \zeta (\alpha - s_n )}}{{\zeta ((n - 1)\sigma )}},\] where $\sigma = s_1 + \cdots + s_n $ and $\zeta (s)$ is the Riemann zeta-function.

The problem of representations by multiple Dirichlet series

The problem of representations by multiple Dirichlet series

Gol'dberg Order and Gol'dberg Type of Entire Functions Represented by Multiple Dirichlet Series

We consider the Hadamard product of the class of entire multiple Dirichlet series in several complex variables having the same sequence of exponents. Our object is to study the nature of Gol'dberg order and Gol’dberg type of these functions. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 29 (2009) 63-70 DOI: http://dx.doi.org/10.3329/ganit.v29i0.8516

Note on Multiple Dirichlet and Multiple Factorial Series

Note on Multiple Dirichlet and Multiple Factorial Series