## Re: [PrimeNumbers] Boland's Distribution of Primes

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• Hello, When g=2762, g^2=7628644, My distribution function, pi(3*g/2)-pi(g/2) ~ pi(g^2)-pi((g-1)^2) predicts 350 primes vs. actual 390 primes, error= -40 or
Message 1 of 14 , Jun 3, 2001
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Hello,

When g=2762, g^2=7628644,
My distribution function,
pi(3*g/2)-pi(g/2) ~ pi(g^2)-pi((g-1)^2)
predicts 350 primes vs. actual 390 primes,
error= -40 or -10.2564%

I conjecture that g=2762 is the highest g for which the
deviation error is greater than 10%.
And I would that someone more skilled than I on this
list can search for a counterexample.

So far I've tested all g up to g=5250, and I had previously
tested all prime g up to ~17,000.

the percentage of the error deviation grows progressively
smaller in amplitude, I began testing a range starting
g=25000 and I would further conjecture that the highest
g with percentage error > 8% will have occurred
prior to g=25000.

It was a theoretical scenario that brought me to test this
function in this neighborhood. I believe my theoretical
argument will make it clear why this phenomenon must
exist within the distribution of prime numbers.

> where the constant c = 3/2*ln(3/2)-1/2*ln(1/2) = 0.95477...

Be aware that my first formulation of
pi(3*g/2)-pi(g/2) ~ pi(g^2)-pi((g-1)^2)
may not be the most exact center for this
"order 1 order 2 codependancy"
within the distribution of primes, but it is close enough
that the percentage error goes to zero with increasing g.

I conjecture that one could consider

pi(3*g/2)-pi(g/2) ~ pi((g-1)^2)-pi((g-2)^2) or
pi(3*g/2)-pi(g/2) ~ pi((g+1)^2)-pi(g^2), for example

and these functions will also yield a percentage error that
goes to zero, maybe slower, maybe faster, somewhere there may
be an exact center (error drops fastest).

the percentage of the error deviation grows progressively
smaller with increasing g.
I began testing a range starting
g=25000 and now I further conjecture that the highest g
for which the percentage error exceeds 8% will have occurred
prior to g=25000.

Here's as far as I got from g=25,000. The
highest percentage error found is < 4% in the tests below.
The sign of the error continues to change frequently
and the percentage of error continues to average
lower & lower.

Can someone please verify some of these numbers for me?

Thanks,

-Dick Boland

Data for g>25000
g g^2 PRED. ACT. ERROR count and %deviation
______________________________________________________
25000 625000000 2476 2431 45 1.8510900863842040312
25001 625050001 2477 2475 2 0.080808080808080808
25002 625100004 2477 2421 56 2.3130937629078893018
25003 625150009 2477 2472 5 0.2022653721682847896
25004 625200016 2477 2465 12 0.4868154158215010141
25005 625250025 2478 2465 13 0.527383367139959432
25006 625300036 2478 2439 39 1.5990159901599015989
25007 625350049 2478 2470 8 0.3238866396761133602
25008 625400064 2478 2390 88 3.68200836820083682
25009 625450081 2478 2503 -25 -0.9988014382740711146
25010 625500100 2478 2489 -11 -0.4419445560466050622
25011 625550121 2478 2480 -2 -0.0806451612903225806
25012 625600144 2479 2466 13 0.5271695052716950526
25013 625650169 2479 2497 -18 -0.7208650380456547856
25014 625700196 2479 2483 -4 -0.1610954490535642368
25015 625750225 2479 2473 6 0.2426202992317023857
25016 625800256 2479 2468 11 0.4457050243111831442
25017 625850289 2479 2428 51 2.1004942339373970345
25018 625900324 2479 2428 51 2.1004942339373970345
25019 625950361 2479 2467 12 0.4864207539521686258
25020 626000400 2480 2466 14 0.5677210056772100567
25021 626050441 2480 2470 10 0.4048582995951417003
25022 626100484 2480 2453 27 1.1006930289441500203
25023 626150529 2480 2487 -7 -0.2814636107760353839
25024 626200576 2479 2493 -14 -0.5615724027276373846
25025 626250625 2480 2429 51 2.0996294771510909839
25026 626300676 2480 2465 15 0.6085192697768762677
25027 626350729 2480 2492 -12 -0.4815409309791332263
25028 626400784 2480 2400 80 3.3333333333333333333
25029 626450841 2480 2516 -36 -1.4308426073131955484
25030 626500900 2480 2512 -32 -1.2738853503184713375
25031 626550961 2480 2520 -40 -1.5873015873015873015
25032 626601024 2481 2490 -9 -0.3614457831325301204
25033 626651089 2482 2471 11 0.4451639012545528126
25034 626701156 2482 2486 -4 -0.1609010458567980691
25035 626751225 2482 2489 -7 -0.2812374447569304941
25036 626801296 2481 2426 55 2.2671063478977741137
25037 626851369 2481 2510 -29 -1.1553784860557768923
25038 626901444 2481 2448 33 1.3480392156862745097
25039 626951521 2481 2456 25 1.0179153094462540716
25040 627001600 2481 2469 12 0.4860267314702308626
25041 627051681 2482 2486 -4 -0.1609010458567980691
25042 627101764 2482 2472 10 0.4045307443365695792
25043 627151849 2482 2477 5 0.2018570851836899474

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• Hello, Has anyone on this list checked out my numbers? Anyone want to know the theory? I need help writing the paper(s), can anyone help me? Nothing worth
Message 2 of 14 , Jun 4, 2001
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Hello,

Has anyone on this list checked out my numbers?
Anyone want to know the theory?
I need help writing the paper(s),
can anyone help me?
Nothing worth writing about here? - I need to
understand why not before wasting my time, or yours.

Thank you

-Dick Boland

--- Dick Boland <richard042@...> wrote:
> Hello,
>
> When g=2762, g^2=7628644,
> My distribution function,
> pi(3*g/2)-pi(g/2) ~ pi(g^2)-pi((g-1)^2)
> predicts 350 primes vs. actual 390 primes,
> error= -40 or -10.2564%
>
> I conjecture that g=2762 is the highest g for which the
> deviation error is greater than 10%.
> And I would that someone more skilled than I on this
> list can search for a counterexample.
>
> So far I've tested all g up to g=5250, and I had previously
> tested all prime g up to ~17,000.
>
> the percentage of the error deviation grows progressively
> smaller in amplitude, I began testing a range starting
> g=25000 and I would further conjecture that the highest
> g with percentage error > 8% will have occurred
> prior to g=25000.
>
> It was a theoretical scenario that brought me to test this
> function in this neighborhood. I believe my theoretical
> argument will make it clear why this phenomenon must
> exist within the distribution of prime numbers.
>
> > where the constant c = 3/2*ln(3/2)-1/2*ln(1/2) = 0.95477...
>
> Be aware that my first formulation of
> pi(3*g/2)-pi(g/2) ~ pi(g^2)-pi((g-1)^2)
> may not be the most exact center for this
> "order 1 order 2 codependancy"
> within the distribution of primes, but it is close enough
> that the percentage error goes to zero with increasing g.
>
> I conjecture that one could consider
>
> pi(3*g/2)-pi(g/2) ~ pi((g-1)^2)-pi((g-2)^2) or
> pi(3*g/2)-pi(g/2) ~ pi((g+1)^2)-pi(g^2), for example
>
> and these functions will also yield a percentage error that
> goes to zero, maybe slower, maybe faster, somewhere there may
> be an exact center (error drops fastest).
>
> the percentage of the error deviation grows progressively
> smaller with increasing g.
> I began testing a range starting
> g=25000 and now I further conjecture that the highest g
> for which the percentage error exceeds 8% will have occurred
> prior to g=25000.
>
> Here's as far as I got from g=25,000. The
> highest percentage error found is < 4% in the tests below.
> The sign of the error continues to change frequently
> and the percentage of error continues to average
> lower & lower.
>
> Can someone please verify some of these numbers for me?
>
> Thanks,
>
> -Dick Boland
>
> Data for g>25000
> g g^2 PRED. ACT. ERROR count and %deviation
> ______________________________________________________
> 25000 625000000 2476 2431 45 1.8510900863842040312
> 25001 625050001 2477 2475 2 0.080808080808080808
> 25002 625100004 2477 2421 56 2.3130937629078893018
> 25003 625150009 2477 2472 5 0.2022653721682847896
> 25004 625200016 2477 2465 12 0.4868154158215010141
> 25005 625250025 2478 2465 13 0.527383367139959432
> 25006 625300036 2478 2439 39 1.5990159901599015989
> 25007 625350049 2478 2470 8 0.3238866396761133602
> 25008 625400064 2478 2390 88 3.68200836820083682
> 25009 625450081 2478 2503 -25 -0.9988014382740711146
> 25010 625500100 2478 2489 -11 -0.4419445560466050622
> 25011 625550121 2478 2480 -2 -0.0806451612903225806
> 25012 625600144 2479 2466 13 0.5271695052716950526
> 25013 625650169 2479 2497 -18 -0.7208650380456547856
> 25014 625700196 2479 2483 -4 -0.1610954490535642368
> 25015 625750225 2479 2473 6 0.2426202992317023857
> 25016 625800256 2479 2468 11 0.4457050243111831442
> 25017 625850289 2479 2428 51 2.1004942339373970345
> 25018 625900324 2479 2428 51 2.1004942339373970345
> 25019 625950361 2479 2467 12 0.4864207539521686258
> 25020 626000400 2480 2466 14 0.5677210056772100567
> 25021 626050441 2480 2470 10 0.4048582995951417003
> 25022 626100484 2480 2453 27 1.1006930289441500203
> 25023 626150529 2480 2487 -7 -0.2814636107760353839
> 25024 626200576 2479 2493 -14 -0.5615724027276373846
> 25025 626250625 2480 2429 51 2.0996294771510909839
> 25026 626300676 2480 2465 15 0.6085192697768762677
> 25027 626350729 2480 2492 -12 -0.4815409309791332263
> 25028 626400784 2480 2400 80 3.3333333333333333333
> 25029 626450841 2480 2516 -36 -1.4308426073131955484
> 25030 626500900 2480 2512 -32 -1.2738853503184713375
> 25031 626550961 2480 2520 -40 -1.5873015873015873015
> 25032 626601024 2481 2490 -9 -0.3614457831325301204
> 25033 626651089 2482 2471 11 0.4451639012545528126
> 25034 626701156 2482 2486 -4 -0.1609010458567980691
> 25035 626751225 2482 2489 -7 -0.2812374447569304941
> 25036 626801296 2481 2426 55 2.2671063478977741137
> 25037 626851369 2481 2510 -29 -1.1553784860557768923
> 25038 626901444 2481 2448 33 1.3480392156862745097
> 25039 626951521 2481 2456 25 1.0179153094462540716
> 25040 627001600 2481 2469 12 0.4860267314702308626
> 25041 627051681 2482 2486 -4 -0.1609010458567980691
> 25042 627101764 2482 2472 10 0.4045307443365695792
> 25043 627151849 2482 2477 5 0.2018570851836899474
>
> __________________________________________________
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• ... We probably all trust you to have got the numerics correct, so checked may not be the right word. They certainly look believable. ... You need more data,
Message 3 of 14 , Jun 4, 2001
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On Mon, 04 June 2001, Dick Boland wrote:
>
> Hello,
>
> Has anyone on this list checked out my numbers?

We probably all trust you to have got the numerics correct, so 'checked' may not be the right word. They certainly look believable.

> Anyone want to know the theory?
> I need help writing the paper(s),
> can anyone help me?
> Nothing worth writing about here? - I need to
> understand why not before wasting my time, or yours.

You need more data, from far higher ranges, before such a prediction makes much sense. When n is small the read deviation may be smaller than the noise.

If you look at www.wolfram.com (the Mathematica website), then I know in the 'Mathematica Book' section, there's am implementation note:
<<<
Prime and PrimePi use sparse caching and sieving. For large n, the Lagarias�Miller�Odlyzko algorithm for PrimePi is
used, based on asymptotic estimates of the density of primes, and is inverted to give Prime.
>>>

Using those names you could try to find the algorithm in question, and using that find some far higher ranges to prove (in the original sense, meaning 'test') your hypothesis.

You might be able to find an online calculator, or Java Applet which does the calculation for you. ('Prime Pi' is the standard name for the function, so it probably a good search string.)

Good luck,
Phil

Mathematics should not have to involve martyrdom;
Support Eric Weisstein, see http://mathworld.wolfram.com
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• ... http://www.math.Princeton.EDU/~arbooker/nthprime.html
Message 4 of 14 , Jun 4, 2001
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Phil Carmody wrote:
> You might be able to find an online calculator
http://www.math.Princeton.EDU/~arbooker/nthprime.html
• Hi, I checked the conjecture by using the nthprime page which David Broadhurst proposed and found for g=10^6, that pi(g^2)-pi((g-1)^2)= 72470 while
Message 5 of 14 , Jun 5, 2001
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Hi,

I checked the conjecture by using the "nthprime"
page which David Broadhurst proposed and found
for
g=10^6, that
pi(g^2)-pi((g-1)^2)= 72470 while
pi(3*g/2)-pi(g/2) = 72617
with a relative difference of 3.4e-3.
Thus, it seems working pretty well. An exact
proof would be most interesting, especially if
providing error bounds.

Ferenc
2,3,5,7,17,23,47,103,107,137,283,313,347,373,...
• pi(x) ~ x/ln(x)*(1+1/ln(x)+O(1/ln(x)^2)) lhs = pi(g^2)-pi((g-1)^2) rhs = pi(3*g/2)-p(g/2) rhs/lhs = 1 + k/log(g) + O(1/ln(g)^2) k = 1 - log(27/4)/2 =
Message 6 of 14 , Jun 5, 2001
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pi(x) ~ x/ln(x)*(1+1/ln(x)+O(1/ln(x)^2))
lhs = pi(g^2)-pi((g-1)^2)
rhs = pi(3*g/2)-p(g/2)
rhs/lhs = 1 + k/log(g) + O(1/ln(g)^2)
k = 1 - log(27/4)/2 = 0.04522874755778077232...

Hence rhs > lhs, at large g, because the
base of Naperian logarithms exceeds sqrt(27/4).
• Let L(g) = pi(g^2) - pi((g-1)^2) R(g) = pi(3*g/2) - pi(g/2) D(g) = R(g) - L(g) where pi(g) is the number of primes not exceeding g. Dick Boland conjectured
Message 7 of 14 , Jun 6, 2001
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Let

L(g) = pi(g^2) - pi((g-1)^2)
R(g) = pi(3*g/2) - pi(g/2)
D(g) = R(g) - L(g)

where pi(g) is the number of primes not exceeding g.

Dick Boland conjectured that D(g) changes
sign an infinite number of times.

On the contrary, I claimed that

k = lim_{g to infty} log(g)^2*D(g)/g = 1 - log(27/4)/2 > 0.

If you replace pi(x) by Riemann's estimator R(x)
(Ribenboim p224) you will find a single sign change
around g=10^4. Superimposed on this upward trend
are sqrt fluctuations from the complex zeros of zeta.
Dick was misled by the fact these can easily buck
the trend for his small g's, around 2.5*10^4.

But for how much longer can this go on?

Already it's getting difficult for g around 10^6,
where a simple sieve of Eratosthenes gave

g R(g) L(g) D(g)
1000000 72617 72450 167 [Pace Ferenc]
999999 72617 72569 48
999998 72617 72340 277
999997 72617 72573 44
999996 72617 72546 71
999995 72617 72381 236
999994 72617 72542 75
999993 72617 72425 192
999992 72617 72548 69
999991 72617 72180 437
999990 72617 72195 422
999989 72617 72561 56
999988 72617 72434 183
999987 72617 72703 -86 [Made it!]
999986 72617 72099 518
999985 72617 72162 455
999984 72616 72378 238
999983 72616 72317 299
999982 72616 72511 105
999981 72616 72371 245
999980 72616 72579 37
999979 72616 72311 305
999978 72616 72352 264
999977 72616 72548 68
999976 72616 72645 -29 [And again!]

These *roughly* agree with a mean k*g/log(g)^2 = 237
and a deviation that is of order sqrt(g/log(g))= 269.

Puzzle: Is there a g>10^7 for which D(g)<0 ?

Here it won't be so easy to
buck the Riemann trend, since
(k*g/log(g)^2)/sqrt(g/log(g)) > 1741/788 > 2.2
• Hello, ... Thanks Phil, Interesting stuff rersulting from this search (besides the algorithm), I will be doing some research to try and put it into context of
Message 8 of 14 , Jun 6, 2001
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Hello,

> Prime and PrimePi use sparse caching and sieving. For large n, the Lagarias�Miller�Odlyzko
> algorithm for PrimePi is
> used, based on asymptotic estimates of the density of primes, and is inverted to give Prime.

Thanks Phil,
Interesting stuff rersulting from this search (besides the algorithm),
I will be doing some research to try and put it into context of my theory
I haven't gotten my hands on the algorithm in a form that I can use,
and it would be good to get some higher data, but it may not be necessary.
The highest prime page is good for some spot checking as Forenc showed,
and still no counterexamples :)

As for Dave's proposition
> pi(x) ~ x/ln(x)*(1+1/ln(x)+O(1/ln(x)^2))
> lhs = pi(g^2)-pi((g-1)^2)
> rhs = pi(3*g/2)-p(g/2)
> rhs/lhs = 1 + k/log(g) + O(1/ln(g)^2)
> k = 1 - log(27/4)/2 = 0.04522874755778077232...
> Hence rhs > lhs, at large g, because the
> base of Naperian logarithms exceeds sqrt(27/4).

I'm not sure that the above proves anything
or if it simply reflects what current
wisdom on the subject would have us believe.
If it's a hard mathematical proof, it would seem to disprove
the conjecture that the sign of the error in my function
changes infinitely often, but not necessarily disprove the
percentage error going to zero.
I need to understand it better, so I have some home work.

I should be able to put something together to share after the weekend.

Thank you,

-Dick Boland

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• ... Yes Dick, that is what I claim: constant sign of difference at sufficiently large g, because you missed a term whose fractional contribution is k/log(g),
Message 9 of 14 , Jun 6, 2001
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Dick Boland wrote:

> it would seem to disprove
> the conjecture that the sign of the error in my function
> changes infinitely often, but not necessarily disprove the
> percentage error going to zero.

Yes Dick, that is what I claim: constant sign
of difference at sufficiently large g, because you
missed a term whose fractional contribution is
k/log(g), which of course goes to zero, relative to
each side, but *dominates* the difference,
when the (roughly!) order 1/sqrt(g) fluctuations die away.

Your less interesting conjecture, that lhs/rhs
goes to unity seems eminently plausible:
both sides are g/log(g) + sub-leading.
No one has *ever* suggested that fluctuations
remain of finite relative size!

My emphasis is on the sub-leading k/log(g) which becomes
(I claim) leading in the relative *difference* R/L-1.
It is masked by fluctuations for g^2 < 10^12,
so you ain't learned nuffin yet :-)
because you stayed at g < 3*10^4.
I believe that k/log(g) dominates fluctuations in R/L-1
*eventually*.

You can use Nth-prime page, for g^2 in [10^12,3*10^13],
like Ferenc, or write an Erato sieve, like me.
Nothing smaller counts, it seems to me.
Wobble masks Riemann for tiny log(g)!

But you are in good company, Andrew Odlyzko
got very worried at g=O(10^22), a few years
ago, when statistical correlations were not
in accord with the *asymptotic* predictions of
the Riemann hypothesis. Then some of Mike Berry's
colleagues in Bristol observed that they could
mock up Andrew's data with random N by N matrices
(Gaussian unitary ensemble, to be technical)
where N is something like log(g)/pi.
So they simulated Odlyzko in tiny amounts of
time (compared to finding the 10^22'nd zero of zeta)
with very modest random matrices (16 by 16 as I recall)
and then easily upped their matrix size to see the onset
of the expected Riemannian behaviour.

Log is a cruel function,
for people interested in asymptotics...
Alain Connes told me that it gave him
the creeps that 10^22 is such a *small* number
when you take its log (and divide by pi as I recall).
You find the 10^22'nd zero of zeta and still
are far away from the prediction of Riemann!

On the other hand, log is good news for prime provers,
with cheap Proths coming at merely log^3 prices.

Best

David
• ... This is a believe: not a proof! The subtlety is that your between squares L(g) = pi(g^2)-pi((g-1)^2) is very intriguing. If pi(g^2) ~ R(g^2)*(1 +/-
Message 10 of 14 , Jun 6, 2001
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PS:

> I believe that k/log(g) dominates fluctuations in R/L-1
> *eventually*.

This is a believe: not a proof!

The subtlety is that your "between squares"
L(g) = pi(g^2)-pi((g-1)^2) is very intriguing.

If
pi(g^2) ~ R(g^2)*(1 +/- O(1/sqrt(g^2))
then naively we get
L(g) ~ g/log(g)*(1 +/- O(1)) [whoops!]

I don't believe that nightmare, since the
ends of the range [(g-1)^2,g^2] are
relatively close together, and hence
tightly correlated.

But you have clearly taken us into
novel (to us) territory, thanks.

David
• A beautifull day Results for Primes mod some numbers up to 10^14 is ready http://beablue.selfip.net/devalco/table_of_primes.htm I checked the results to P mod
Message 11 of 14 , Jan 23, 2010
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A beautifull day

Results for Primes mod some numbers up to 10^14 is ready
http://beablue.selfip.net/devalco/table_of_primes.htm
I checked the results to P mod 4 = 3 and P mod 4 = 1
concerning the existing table.

I used a sieve of Eratosthenes with a Heap-construction for collecting the primes and Help-array in the first level Cache for sieving the primes.

Program under
http://beablue.selfip.net/devalco/sieb_des_eratosthenes.htm

Runtime of the program is 7*14 days, i distributed the work on 7 nodes.

There can be made some improvements using assembler.

I would like to expand the tables with the distribution of Primes up to 10^15 or 10^16.

Is there a chance to calculate the programs in a grid or cluster.
A connection between the node is not necessary.

Besides the results could be usefull for physic or biologic research

Nice Greetings from the primes
Bernhard

http://www.devalco.de
• ... 10^15 or 10^16. http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1001&L=nmbrthry&T=0&X=14ADB57FE44944E3D4&P=327 Best regards, Andrey
Message 12 of 14 , Jan 23, 2010
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> I would like to expand the tables with the distribution of Primes up to
10^15 or 10^16.