Asymptotic behavior of Alternating Ordinary Dirichlet Series remainder 2nd
Luca
Hi,
I am posting this New Topic in an effort to verify whether it is an already
known result that the n^th remainder, R_n(s) of an Alternating Ordinary
Dirichelet Series (meaning: 1 - 2^-s + 3^-s - ..., s being a complex number)
is
Big Theta(n^(-Re(s))), as n->oo.
Now, by means of standard analysis it is relatively easy to show that R_n(s)
is
Big O(n^(-Re(s))), as n->oo.
Big Theta represents instead a stricter asymptotic condition than Big O.
If needed, definitions of Big O and Big Theta can be found at
en.wikipedia.org/wiki/Big_O_notation.
I am asking this question because I have quite accidentally come across a
geometric proof of this result, but not being a true expert on the subject
(myself I am ot a professional mathematician, but simply an industrial
physicist who came across this result quite accidentally, while studying
some elementary prime number theory to get more background knowledge for a
cryptography application I was developing) I would be grateful to other
interested scholars if they could comment about the possible novelty of this
finding, or whether they find it of any interest at all ...
In any case, if ever interested, the details of said geometric proof can be
found in a manuscript (it is the proof of Theorem 1) I have published on the
arXiv arxiv.org/abs/0907.2426 .
Mind you, in that manuscript I then use said Big Theta result to derive an
Hypothesis (also possibly novel) equivalent to the Riemann Hypothesis, but
here my question relates only to said Big Theta asymptotic behavior.
Thank you for any comments you may have,
Luca
P.S.: as a quick summary, the abstract of the above mentioned manuscript
reads as follows
ABSTRACT
For any s in C with Re(s)>0, denote by S_n (s) the n^th partial sum of the
alternating Dirichlet series 1 - 2^-s + 3^-s - ... We first show that S_n
(s) =/= 0 for all n greater than some index N(s). Denoting by D = { s in C:
0< Re(s) < = 1/2 } the open left half of the critical strip, define for all
s in D and n > N(s) the ratio P_n (s) = S_n (1-s) / S_n (s). We then prove
that the limit L(s) = Lim P_n (s) (N(s) < n --> oo) exists at every point s
of the domain D. Finally, we show that the function L(s) is continuous on D
if and only if the Riemann Hypothesis is true.
An effective way to visualize in one's mind the above result refers to fig.2
of the manuscript:
The limit function L(s) exists regardless of whether or not the the RH is
true. If the RH is true, then L(s) is a continuous function and its modulus
is the function plotted in Fig. 2. If the RH is not true, then L(s) still
coincides with the function whose modulus is plotted in Fig. 2, excepts at
the locations of the off-the-critical-line zeros, where it will feature
discontinuities L(s) = 0.
Fig. 9 gives then an idea of how said hypothetical off-the-critical-line
zeros would affect the convergence pattern of said ratios Pn(s).