scholarly journals Note on the region of overconvergence of Dirichlet series with Ostrowski gaps

1966 ◽  
Vol 7 (4) ◽  
pp. 169-173 ◽  
Author(s):  
J. P. Earl ◽  
J. R. Shackell
Keyword(s):  
Power Series ◽  
Dirichlet Series ◽  
Partial Sums ◽  
Direct Treatment

The main object of this note is to show that a proof given by A. J. Macintyre [2] of a result on the overconvergence of partial sums of power series works more easily in the context of Dirichlet series. Applying this observation to the particular Dirichlet series Σane−ns, we can remove certain restrictions which Macintyre finds necessary in the direct treatment of power series.

A Multiplicative Analogue of Schur's Tauberian Theorem

2003 ◽  
Vol 46 (3) ◽  
pp. 473-480 ◽  
Author(s):  
Karen Yeats
Keyword(s):  
Power Series ◽  
Asymptotic Behaviour ◽  
Dirichlet Series ◽  
Tauberian Theorem ◽  
Partial Sums ◽  
Regularly Varying Functions ◽  
Regularly Varying

AbstractA theorem concerning the asymptotic behaviour of partial sums of the coefficients of products of Dirichlet series is proved using properties of regularly varying functions. This theorem is a multiplicative analogue of Schur's Tauberian theorem for power series.

scholarly journals The power series coefficients of functions defined Dirichlet series

1961 ◽  
Vol 5 (1) ◽  
pp. 43-44 ◽  
Author(s):  
W. E. Briggs ◽  
R. G. Buschman
Keyword(s):  
Power Series ◽  
Dirichlet Series

scholarly journals Variant of the truncated Perron formula and primes in polynomial sets

2019 ◽  
Vol 16 (02) ◽  
pp. 309-323
Author(s):  
D. S. Ramana ◽  
O. Ramaré
Keyword(s):  
Dirichlet Series ◽  
Riemann Hypothesis ◽  
Partial Sums ◽  
Generalized Riemann Hypothesis ◽  
Perron’S Formula ◽  
Polynomial Formula

We show under the Generalized Riemann Hypothesis that for every non-constant integer-valued polynomial [Formula: see text], for every [Formula: see text], and almost every prime [Formula: see text] in [Formula: see text], the number of primes from the interval [Formula: see text] that are values of [Formula: see text] modulo [Formula: see text] is the expected one, provided [Formula: see text] is not more than [Formula: see text]. We obtain this via a variant of the classical truncated Perron’s formula for the partial sums of the coefficients of a Dirichlet series.

scholarly journals On (J, pn)-summability of Fourier series

1972 ◽  
Vol 18 (1) ◽  
pp. 13-17
Author(s):  
F. M. Khan
Keyword(s):  
Fourier Series ◽  
Power Series ◽  
Radius Of Convergence ◽  
Partial Sums ◽  
Summability Of Fourier Series ◽  
Image Position

Let pn>0 be such that pn diverges, and the radius of convergence of the power seriesis 1. Given any series σan with partial sums sn, we shall use the notationand

The zeros of the partial sums of power series

1963 ◽  
Vol 30 (4) ◽  
pp. 533-540 ◽  
Author(s):  
T. Ganelius
Keyword(s):  
Power Series ◽  
Partial Sums

scholarly journals On partial sums of Lagrange's series with application to the theory of queues

1963 ◽  
Vol 3 (4) ◽  
pp. 488-490 ◽  
Author(s):  
P. D. Finch
Keyword(s):  
Power Series ◽  
Partial Sums ◽  
Lagrange's Series ◽  
Lagrange's Theorem

Lagrange's theorem on the reversion of power series may be stated in the following form (e.g. Whittker and Watson [3]).

RESTRICTED UNIVERSALITY OF POWER SERIES

2001 ◽  
Vol 33 (5) ◽  
pp. 543-552 ◽  
Author(s):  
JEAN-PIERRE KAHANE ◽  
ANTONIOS D. MELAS
Keyword(s):  
Power Series ◽  
Finite Union ◽  
Radius Of Convergence ◽  
Nonempty Interior ◽  
Partial Sums ◽  
Universal Approximation ◽  
Compact Set ◽  
Approximation Properties

We prove the existence of a power series having radius of convergence 0, whose partial sums have universal approximation properties on any compact set with connected complement that is contained in a finite union of circles centred at 0 and having rational radii, but do not have such properties on any compact set with nonempty interior. This relates to a theorem of A. I. Seleznev.

Sign in / Sign up

Export Citation Format

Share Document