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• ... Since zeta(n) = 1 + sum_{k=2}^oo k^{-n} = 1 + 2^{-n}(1 + (2/3)^n + (2/4)^n + (2/5)^n + ...) while zeta(x) ~ 1/(x-1) as x - 1, zeta(zeta(n)) ~ 2^n. This
Oct 1, 2005 1 of 7
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--- In mathforfun@yahoogroups.com, "cino hilliard" <hillcino368@h...>
wrote:
> Consider the sequence,
>
> [nearly doubling integers]
>
> which is the Integer part of Zeta(Zeta(n)), n=2,3,..
>
> Conjecture: Zeta(Zeta(n))/Zeta(Zeta(n+1)) -> 1/2 as n -> oo
>
> Can someone prove this?
>

Since

zeta(n)
= 1 + sum_{k=2}^oo k^{-n}
= 1 + 2^{-n}(1 + (2/3)^n + (2/4)^n + (2/5)^n + ...)

while zeta(x) ~ 1/(x-1) as x -> 1,

zeta(zeta(n)) ~ 2^n.

This asymptotic estimate may not be close enough to give the correct
integer part of zeta(zeta(n)), but it's certainly enough to imply the
truth of your observation.

Regards,

• Hi Adh_math, ... I don t follow this. Maybe you can do an example. ... Indeed, zeta(zeta(50)) = 112589990(5076842.61671.. 2^50 = 112589990(6842624
Oct 3, 2005 1 of 7
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>To: mathforfun@yahoogroups.com
>Subject: [MATH for FUN] Re: Zeta(Zeta(n))
>Date: Sat, 01 Oct 2005 12:30:12 -0000
>
>--- In mathforfun@yahoogroups.com, "cino hilliard" <hillcino368@h...>
>wrote:
> > Consider the sequence,
> >
> > [nearly doubling integers]
> >
> > which is the Integer part of Zeta(Zeta(n)), n=2,3,..
> >
> > Conjecture: Zeta(Zeta(n))/Zeta(Zeta(n+1)) -> 1/2 as n -> oo
> >
> > Can someone prove this?
> >
>
>Since
>
> zeta(n)
> = 1 + sum_{k=2}^oo k^{-n}
> = 1 + 2^{-n}(1 + (2/3)^n + (2/4)^n + (2/5)^n + ...)
>
>while zeta(x) ~ 1/(x-1) as x -> 1,
I don't follow this. Maybe you can do an example.

> zeta(zeta(n)) ~ 2^n.
Indeed,
zeta(zeta(50)) = 112589990(5076842.61671..
2^50 = 112589990(6842624
zeta(zeta(100))= 1267650600228229(398378720795167.6352446..
2^100 = 1267650600228229(401496703205376
zeta(zeta(200) =
1606938044258990275541962092341(2592800388683416665871349746.992..
2^200 = =
16069380442589902755419620923411(62602522202993782792835301376

Certainly then 2^n = 1/2*2^(n+1)
I should have observed that too.

Cino.
• ... ... Hi Cino, The behavior of zeta near 1 is a Standard Fact. (Zeta has a meromorphic continuation to the plane, with a simple pole of
Oct 4, 2005 1 of 7
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--- In mathforfun@yahoogroups.com, "cino hilliard" <hillcino368@h...>
wrote:
>
> >
> >--- In mathforfun@yahoogroups.com, "cino hilliard"
<hillcino368@h...>
> >wrote:
> > > Consider the sequence,
> > >
> > > [nearly doubling integers]
> > >
> > > which is the Integer part of Zeta(Zeta(n)), n=2,3,..
> > >
> > > Conjecture: Zeta(Zeta(n))/Zeta(Zeta(n+1)) -> 1/2 as n -> oo
> > >
> > > Can someone prove this?
> > >
> >
> >Since
> >
> > zeta(n)
> > = 1 + sum_{k=2}^oo k^{-n}
> > = 1 + 2^{-n}(1 + (2/3)^n + (2/4)^n + (2/5)^n + ...)
> >
> >while zeta(x) ~ 1/(x-1) as x -> 1,
>
> I don't follow this. Maybe you can do an example.
>
Hi Cino,

The behavior of zeta near 1 is a Standard Fact. (Zeta has a
meromorphic continuation to the plane, with a simple pole of residue 1
at z=1. :) The only proofs I know use technology (product expansions,
complex line integrals), and can be found in Ahlfors' Complex
Analysis. For present purposes, we don't need the residue (a.k.a. the
coefficient k if zeta(x) ~ k/(x-1) near 1), just the order of the
pole, but off the top of my head I don't see an easy (low-tech) proof
that the pole is simple...

On the subject, however:
A peasant from Gdansk was travelling to America to see his relatives.
It was his first time aboard an airplane, and a 747 at that. Over the
Atlantic, the plane was hijacked. The pilots were injured in the
scuffle, and a masked man grabbed the peasant by the sleeve, shoved
him into the cockpit, and at gunpoint told him to fly the plane. "But
sir", stammered the peasant, gazing in awe at the controls, "this is a
complex plane and I am just a simple Pole."

Regards,
• methinks adh just revealed that s/he s in Europe. Correct? ... relatives. ... In Europe it s spelled travelling whereas in America it s spelled traveling
Oct 6, 2005 1 of 7
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methinks adh just revealed that s/he's in Europe. Correct?
>
> On the subject, however:
> A peasant from Gdansk was ***travelling*** to America to see his
relatives.
> It was his first time aboard an airplane, and a 747 at that. Over the
> Atlantic, the plane was hijacked. The pilots were injured in the
> scuffle, and a masked man grabbed the peasant by the sleeve, shoved
> him into the cockpit, and at gunpoint told him to fly the plane. "But
> sir", stammered the peasant, gazing in awe at the controls, "this is a
> complex plane and I am just a simple Pole."
>
> Regards,

In Europe it's spelled 'travelling' whereas in America it's
spelled 'traveling'
• ... Actually not, just a poor typist... :)
Oct 8, 2005 1 of 7
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--- In mathforfun@yahoogroups.com, bqllpd <no_reply@y...> wrote:
>
> > A peasant from Gdansk was ***travelling*** to America to see his
> > relatives.
> methinks adh just revealed that s/he's in Europe. Correct?
>

Actually not, just a poor typist... :)
• ... Actually I think the single l in American traveling is another example of only in America It is certainly travelling in Australia and New Zealand, and
Oct 9, 2005 1 of 7
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--- In mathforfun@yahoogroups.com, bqllpd <no_reply@y...> wrote:

Actually I think the single l in American traveling is another
example of "only in America"

It is certainly travelling in Australia and New Zealand, and any
other country I have visited [though I didn't look too carefully in

Peter

>
> methinks adh just revealed that s/he's in Europe. Correct?
> >
>
> In Europe it's spelled 'travelling' whereas in America it's
> spelled 'traveling'
>
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