Displaying similar documents to “On the number of subgroups of finite abelian groups”

On the mean value of the generalized Dirichlet L -functions

Rong Ma, Yuan Yi, Yulong Zhang (2010)

Czechoslovak Mathematical Journal

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Let q 3 be an integer, let χ denote a Dirichlet character modulo q . For any real number a 0 we define the generalized Dirichlet L -functions L ( s , χ , a ) = n = 1 χ ( n ) ( n + a ) s , where s = σ + i t with σ > 1 and t both real. They can be extended to all s by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet L -functions especially for s = 1 and s = 1 2 + i t , and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.

On the lower order ( R ) of an entire Dirichlet series

P. K. Jain, D. R. Jain (1974)

Annales de l'institut Fourier

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The estimations of lower order ( R ) λ in terms of the sequences { a n } and { λ n } for an entire Dirichlet series f ( s ) = n = 1 a n e s λ n , have been obtained, namely : λ = max { λ n p } lim inf p λ n p log λ n p - 1 log | a n p | - 1 = max { λ n p } lim inf p ( λ n p - λ n p - 1 ) log λ n p - 1 log | a n p - 1 | a n p | . One of these estimations improves considerably the estimations earlier obtained by Rahman (Quart. J. Math. Oxford, (2), 7, 96-99 (1956)) and Juneja and Singh (Math. Ann., 184(1969), 25-29 ).

Limit theorem in the space of continuous functions for the Dirichlet polynomial related with the Riemann zeta-funtion

Antanas Laurinčikas (1996)

Journal de théorie des nombres de Bordeaux

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A limit theorem in the space of continuous functions for the Dirichlet polynomial m T d κ T ( m ) m σ T + i t where d κ T ( m ) denote the coefficients of the Dirichlet series expansion of the function ζ κ T ( s ) in the half-plane σ > 1 κ T = ( 2 - 1 log l T ) - 1 2 , σ T = 1 2 + 1 n 2 l T l T and l T > 0 , l T 1n T and l T as T , is proved.

On the value distribution of a class of arithmetic functions

Werner Georg Nowak (1996)

Commentationes Mathematicae Universitatis Carolinae

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This article deals with the value distribution of multiplicative prime-independent arithmetic functions ( α ( n ) ) with α ( n ) = 1 if n is N -free ( N 2 a fixed integer), α ( n ) > 1 else, and α ( 2 n ) . An asymptotic result is established with an error term probably definitive on the basis of the present knowledge about the zeros of the zeta-function. Applications to the enumerative functions of Abelian groups and of semisimple rings of given finite order are discussed.