Talk:Dirichlet series

Vague

a Dirichlet series is a series taken over the inverse powers of the integers.

The above seems horribly vague. "Powers"—plural—seems to imply the values of the exponents will be different. If all I had to go by was the sentence above, I would have no way to tell what is and what is not a Dirichlet series. Michael Hardy 22:41, 17 May 2006 (UTC)[reply]

Shifted index

sorry..for the inconvinience, but if:

$F(s)=\sum _{n=1}^{\infty }f(n)n^{-s}$

then what could we say about G(s) so:

$G(s)=\sum _{n=1}^{\infty }f(n+k)n^{-s}$

where k is a positive integer, is there any relationship between G(s) and F(s) ?? --85.85.100.144 21:36, 21 February 2007 (UTC)[reply]

I don't see any connexion other than what you have written. Eric Kvaalen (talk) 07:37, 20 August 2014 (UTC)[reply]

Convergence

In connexion with the sentence "In general the abscissa of convergence of a Dirichlet series is the intercept on the real axis of the vertical line in the complex plane, such that there is convergence to the right of it, and divergence to the left" User:LokiClock put a "clarification needed" tag and asks "How do any of them converge absolutely on one half-plane, uniformly on the other if they all diverge to the left of the abscissa?" Well, it's not saying that it converges absolutely on one half-plane and uniformly on the other. It's just saying that you have convergence on the right and divergence on the left. It goes on to say that the planes of absolute convergence and of uniform convergence may be different, but both of these will be on the right. For the Riemann zeta function (or series), one has absolute convergence so long as Re(s)>1, but the convergence is not uniform in this half-plane. It seems to me that convergence is uniform in any half-plane Re(s) ≥ b > 1, but none of those includes all other half-planes of uniform convergence, so one cannot talk about the half-plane of uniform convergence. Eric Kvaalen (talk) 07:37, 20 August 2014 (UTC)[reply]

Comma in definition

Is the trailing comma in the definition:

\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}},

significant or simply someone's pedantic idea of "correct" punctuation? If the latter, why is it inside the <math> tag; indeed, why is it there at all? TIA HAND —Phil | Talk 13:49, 11 May 2016 (UTC)[reply]

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