- The Riemann Conjecture has not been proved wrong yet.

The Clay Institute are still looking after the $1,000,000 for me ;-)

I do remember reading that someone had proved that a Littlewood

Conjecture _OR_ Riemann's Hypothesis is FALSE.

I think this must have been on the web and I have half a memory that

the title misused the word "paradox".

I have tried searching the web and my browser's history but I can't

find it again :-(

Does anyone know the reference or can suggest anything to tighten

my searches?

Cheers,

Paul Landon

Andrey Kulsha wrote:

[snip]

> We see: at first Gauss conjectured that func=1, i.e. P(x)~Li(x). But

> this was proven to be wrong.

>

> Then Riemann conjectured func=zeta, and now we know it's wrong too (a

> contradiction with Littlewood's result).

>

> I still have no any ideas about func(n+1).

>

> Maybe somebody know about any papers/books discussing this problems?

>

> Great thanks,

>

> Andrey

>

> Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com

> The Prime Pages : http://www.primepages.org

>

>

>

> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ - Andrey Kulsha wrote:

> I.J. Good and R.F. Churchhouse, The Riemann hypothesis and

My word, that is a fascinating paper, Andrey.

> pseudorandom features of the Moebius sequence,

> Math. Comp. 22(1968), 857-861.

> Note that conjecture is also based on statistics, but

> it's too strong. I've obtained a similar result,

> but I write "O[]" and "+eps"

First of all it used up most of the spare computing power

in UK at the time, being run in the background at

the Chilton Atlas lab. However some of the results were lost

because of "a single supervisor fault". I bet that

poor person was made to feel guilty.

Secondly it anticipates a disproof of Merten's conjecture.

These authors clearly took log(log(x)) seriously.

They discount the evidence at the time (up to 2 million zeros)

for the Riemann hypothesis, on the grounds that a failure

would involve loglog asymptotics. They quote Littlewood

as saying "there is no imaginable reason why it should be true"

and then go on to try to give a "reason". They have wise

things to say about the failure of the Moebius sequence to

be truly random. Their sqrt(12)/pi conjecture came from

taking out multiples of squares, which give a

1/zeta(2)=6/pi^2. I guess that is too simplistic, but

the paper gives a good idea of how folk were grappling

with loglog asymptotics when computing power became available

in the late 60's.

Moore's log law looks mean when you take *its* log.

If log of processing power increases linearly with time,

then our loglog progress clearly decelerates.

There is perhaps less to be learnt in the present decade

than was learnt in the 60's!

Andrey does well to investigate

how much we have learnt in the last 35 years,

on a loglog scale, when we progressed

by perhaps 40% from exp(exp(2.7)) to exp(exp(3.9))

in terms of zeta zeros. The largest prime grew from

exp(exp(9)) in 1963 to exp(exp(15)), but that gives

us no statistical information. More pertinent is

that Meissel could make good sense of pi(exp(exp(3)))

in the 1880's, yet today we do not know pi(exp(exp(4))).

Modest progress indeed!

David