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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=20
Big Theta(n^(-Re(s))), as n->oo.=20
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=20
P.S.: as a quick summary, the abstract of the above mentioned manuscript =
reads as follows=20
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) =3D/=3D 0 for all n greater than some index N(s). Denoting =
by D =3D { s in C: 0< Re(s) < =3D 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) =3D =
S_n (1-s) / S_n (s). We then prove that the limit L(s) =3D 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) =3D 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).
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<DIV><FONT size=3D2 face=3DArial>Hi,<BR>I am posting this New Topic in =
an effort to=20
verify whether it is an already known result that the n^th remainder, =
R_n(s) of=20
an Alternating Ordinary Dirichelet Series (meaning: 1 - 2^-s + 3^-s - =
..., s=20
being a complex number) is </FONT></DIV>
<DIV><FONT size=3D2 face=3DArial><BR>Big Theta(n^(-Re(s))), as n->oo. =
</FONT></DIV><FONT size=3D2 face=3DArial>
<DIV><BR>Now, by means of standard analysis it is relatively easy to =
show that=20
R_n(s) is</DIV>
<DIV><BR>Big O(n^(-Re(s))), as n->oo.</DIV>
<DIV><BR>Big Theta represents instead a stricter asymptotic condition =
than Big=20
O.<BR>If needed, definitions of Big O and Big Theta can be found =
at <A=20
href=3D"http://en.wikipedia.org/wiki/Big_O_notation">en.wikipedia.org/wik=
i/Big_O_notation</A>.</FONT></DIV>
<DIV><FONT size=3D2 face=3DArial></FONT> </DIV>
<DIV><FONT size=3D2 face=3DArial>I am asking this question because I =
have quite=20
accidentally come across a geometric proof of this result, but not being =
a true=20
expert on the subject (myself I am ot a professional mathematician, but =
simply=20
an industrial physicist who came across this result quite accidentally, =
while=20
studying some elementary prime number theory to get more background =
knowledge=20
for a cryptography application I was developing) I would be grateful to =
other=20
interested scholars if they could comment about the possible novelty of =
this=20
finding, or whether they find it of any interest at all ...</FONT></DIV>
<DIV> </DIV>
<DIV><FONT size=3D2 face=3DArial>In any case, if ever interested, the =
details of=20
said geometric proof can be found in a manuscript (it is the proof of =
Theorem 1)=20
I have published on the arXiv <A=20
href=3D"http://arxiv.org/abs/0907.2426">arxiv.org/abs/0907.2426</A> =
.</FONT></DIV>
<DIV> </DIV>
<DIV><FONT size=3D2 face=3DArial>Mind you, in that manuscript I then use =
said Big=20
Theta result to derive an Hypothesis (also possibly novel) equivalent to =
the=20
Riemann Hypothesis, but here my question relates only to said Big Theta=20
asymptotic behavior.</FONT></DIV>
<DIV> </DIV>
<DIV><FONT size=3D2 face=3DArial>Thank you for any comments you may=20
have,</FONT></DIV>
<DIV> </DIV>
<DIV><FONT size=3D2 face=3DArial>Luca </FONT></DIV>
<DIV> </DIV>
<DIV><FONT size=3D2 face=3DArial>P.S.: as a quick summary, the abstract =
of the above=20
mentioned manuscript reads as follows <BR>ABSTRACT<BR>For any s in C =
with=20
Re(s)>0, denote by S_n (s) the n^th partial sum of the alternating =
Dirichlet=20
series 1 - 2^-s + 3^-s - ... We first show that S_n (s) =3D/=3D 0 for =
all n greater=20
than some index N(s). Denoting by D =3D { s in C: 0< Re(s) < =3D =
1/2 } the=20
open left half of the critical strip, define for all s in D and n > =
N(s) the=20
ratio P_n (s) =3D S_n (1-s) / S_n (s). We then prove that the limit L(s) =
=3D Lim P_n=20
(s) (N(s) < n --> oo) exists at every point s of the domain D. =
Finally, we=20
show that the function L(s) is continuous on D if and only if the =
Riemann=20
Hypothesis is true.</FONT></DIV>
<DIV> </DIV>
<DIV><FONT size=3D2 face=3DArial>An effective way to visualize in one's =
mind the=20
above result refers to fig.2 of the manuscript:<BR>The limit function =
L(s)=20
exists regardless of whether or not the the RH is true. If the RH is =
true, then=20
L(s) is a continuous function and its modulus is the function plotted in =
Fig. 2.=20
If the RH is not true, then L(s) still coincides with the function whose =
modulus=20
is plotted in Fig. 2, excepts at the locations of the =
off-the-critical-line=20
zeros, where it will feature discontinuities L(s) =3D 0.</FONT></DIV>
<DIV> </DIV>
<DIV><FONT size=3D2 face=3DArial>Fig. 9 gives then an idea of how said =
hypothetical=20
off-the-critical-line zeros would affect the convergence pattern of said =
ratios=20
Pn(s).</FONT></DIV>
<DIV><FONT size=3D2 face=3DArial></FONT> </DIV>
<DIV><FONT size=3D2 face=3DArial> </FONT></DIV></BODY></HTML>
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