Asymptotic behavior of Alternating Ordinary Dirichlet Series remainder.
Luca
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 http://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 (http://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).
David Bernier
> 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.
I have read a little bit about the Riemann zeta function and ways to
compute it. One of the simplest methods to use is Euler-Maclaurin
summation, and Harold Edwards' book "The Riemann Zeta Function"
gives many examples of this method for various functions, including
the Gamma function and the Riemann zeta function.
For the zeta function, n^(-s) gives rise to f(x) = x^(-s) , which
can be integrated with respect to x.
[Formula (1), Section 6.4 of Edwards ].
There may be a way to apply Euler-Maclaurin summation also
to the Alternating Ordinary Dirichlet Series
1 - 2^(-s) + 3^(-s) - 4^(-s) + ... - ... + ...
One idea would be to set
g(x) := (-1)^(x-1) x^(-s) , so as to reproduce the alternating sign
pattern when replacing 'x' by 'n'.
I understand how to apply Euler-Maclaurin summation, but not
how or why it works (the proofs).
David Bernier
David Bernier
> Luca wrote:
>> 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.
>
> I have read a little bit about the Riemann zeta function and ways to
> compute it. One of the simplest methods to use is Euler-Maclaurin
> summation, and Harold Edwards' book "The Riemann Zeta Function"
> gives many examples of this method for various functions, including
> the Gamma function and the Riemann zeta function.
>
> For the zeta function, n^(-s) gives rise to f(x) = x^(-s) , which
> can be integrated with respect to x.
> [Formula (1), Section 6.4 of Edwards ].
>
> There may be a way to apply Euler-Maclaurin summation also
> to the Alternating Ordinary Dirichlet Series
> 1 - 2^(-s) + 3^(-s) - 4^(-s) + ... - ... + ...
I looked at articles by P. Borwein and others, and also
one by Xavier Gourdon and Pascal Sebah.
The 2003 paper by Gourdon and Sebah refers to a 1995 preprint
by P. Borwein and others about "acceleration of convergence"
for computing Dirichlet eta, eta(s), and improved algorithms
compared to Euler-Maclaurin summation for high-accuracy evaluation
of zeta(s) using a high-accuracy computation of eta(s).
The Gourdon and Sebah paper, in the file
zetaevaluations.pdf can be downloaded from here:
< http://numbers.computation.free.fr/Constants/Miscellaneous/ >
In Section 1.1, pages 1 and 2, they give the general
Euler-Maclaurin formula for zeta(s) where
- q determines the number of "Bernoulli number"-related
small correction terms.
- N is the largest index in the zeta(.) partial sum.
In practice one needs about N >= Im(s), where
we assume Im(s) > 0.
- s = sigma + it , so sigma = 1/2 for the critical line.
I mention page 2 because they call the error term
E_{2q} (s) [ here, the dependence on 'N' is
not explicit, but it does depend on N].
They give on page 2 a bound
| E_{2q} (s) | < ... , which they attribute to Backlund;
H. Edwards cites Backlund, 1916 as a source for this bound.
If we fix q = 4, say, and s is fixed, the B_2 term is included and
lim_{N -> oo} | E_{4} (s) | = 0.
On the other hand, there's a correction term + N^(1-s)/(s-1) .
If sigma < 1, then | N^(1-s)| = |N^(1-sigma)| and as
N -> oo, |N^(1-sigma)| -> oo . So for example, if sigma = 1/2,
then the partial sums sum_{n = 1 ... N} n^(-1/2 - it) are
unbounded.
===
Section 1.2 of Gourdon and Sebah's survey paper mentions
eta computations as a means to compute zeta accurately.
They refer to Borwein, Borwein and Jakinovski [1995] for
improved algorithms to compute eta(.). They refer to
Cohen, F. Rodriguez Villegas, D. Zagier [1991] regarding
acceleration of convergence in alternating series.
In the Dirichlet series for eta(s), the n'th term is
+/- n^(-s) with absolute value n^(-sigma).
So following your notation,
| R_n(s) | >= 1/2 n^(-sigma) = 1/2 n^(-Re(s)) if n is large enough.
| R_n(s) | <= K n^(-Re(s)) for some fixed K> 0 seems plausible
to me, at least if
1- epsilon < Re(s) < 1, small epsilon.
This can be abbreviated R_n(s) = O(n^(-sigma)).
You wrote originally:
> 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.
You might well have a geometric proof of this.
I didn't look at everything in your article.
For sigma < 1/2, the size of the summands goes
down more slowly as n increases [than if sigma = 0.8, e.g.].
David Bernier
David Bernier
> David Bernier wrote:
>> Luca wrote:
>>> 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.
I tried to find a reference for the O(.) bound, and
I found that Dirichlet eta and Riemann zeta are
special cases of the Lerch zeta function L(lambda, alpha, s),
where lambda and alpha are parameters that specify a unique
zeta-like function, via a Dirichlet series.
Antanas Laurincikas and Ramunas Garunkstis have
written a book about the Lerch zeta function.
Garunkstis has since then written a short paper about
partial sum approximations to the Lerch zeta function:
R. Garunkstis
"Approximation of the Lerch zeta-function",
Lithuanian Math. Journal, 2004.
Cf.:
< http://www.mif.vu.lt/~garunkstis/ >
paper number 26 in PDF format.
In the Lerch zeta function L(lambda, alpha, s),
if we set lambda = 1/2 and alpha = 1, we get:
L(1/2, 1, s) = sum_{m=0, .. oo} exp(pi*i*m)/((m+1)^s) or
L(1/2, 1, s) = sum_{n = 1 ... oo} (-1)^(n+1) n^(-s).
So L(1/2, 1, s) = eta(s) if Re(s) > 1.
Garunkstis writes that the Lerch zeta function can be analytically
continued to the whole complex plane except possibly at s = 1.
He gives as a reference the book
"The Lerch Zeta function", by Laurincikas and Garunkstis.
Cf.:
< http://www.amazon.com/gp/product/1402010141/ >
He refers to Chapter 3 of his book with Laurincikas
for the result
L(lambda, alpha, s) = sum_{m=0 ... N} exp(pi*i*m)/((m+1)^s) + O(
N^(-sigma)),
where sigma = Re(s).
This is subject to 0<lambda<1 and sigma>0.
[ By limiting lambda to (0, 1) , we will exclude
Riemann zeta, which makes sense since I believe
the partial sums diverge in general for
Riemann zeta ].
For the alternating Dirichlet sum, we let lambda = 1/2 and alpha = 1,
to obtain eta(.) i.e. L(1/2, 1, .) as above.
David Bernier
Luca
many thanks for your suggestions, you are a mine of references!
My apologies for taking so long to reply (I thought I had selected "automatic notifications" by e-mail, but apparently I am still having troubles with it).
So, If I can summarise the results of your search, you were also able to confirm that said remainder is already known to be Big-O of 1/n^s, but that the stricter condition Big-Theta of 1/n^s would instead appear to represent a novel finding.
I am currently thinking to try to get said result published on a peer review Math Journal.
The corresponding manuscript would be much shorter than the one published on ArXiv, and strictly focused on Theorem 1 and its Corollaries.
However, not being a professional mathematicians I have no idea about which Journal would be suitable.
Would you have any advice about it ? Mind you, as you might have noticed, I am certainly not capable to write a paper in the most perfect math style and rigour. So, I would rather try with Journals known to be not too "obsessed" with absolute perfection, if any exist ...
Thank you for your kind attention
Luca
David Bernier
> David,
> many thanks for your suggestions, you are a mine of references!
> My apologies for taking so long to reply (I thought I had selected "automatic notifications" by e-mail, but apparently I am still having troubles with it).
>
> So, If I can summarise the results of your search, you were also able to confirm that said remainder is already known to be Big-O of 1/n^s, but that the stricter condition Big-Theta of 1/n^s would instead appear to represent a novel finding.
I don't really feel competent enough to confirm the Big-Theta of 1/n^s
statement. It might well be true. What I did find was a paper
citing the book on the Lerch zeta function about the
Big-O of 1/n^s result.
>
> I am currently thinking to try to get said result published on a peer review Math Journal.
> The corresponding manuscript would be much shorter than the one published on ArXiv, and strictly focused on Theorem 1 and its Corollaries.
> However, not being a professional mathematicians I have no idea about which Journal would be suitable.
> Would you have any advice about it ? Mind you, as you might have noticed, I am certainly not capable to write a paper in the most perfect math style and rigour. So, I would rather try with Journals known to be not too "obsessed" with absolute perfection, if any exist ...
I don't feel competent enough in analytical number theory to give
advice about publication.
You could look at the pages of the Number Theory Web:
< http://www.numbertheory.org/ntw/web.html > .
> Thank you for your kind attention
>
> Luca
Best wishes,
David