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a pie in the sky, via the number pi

47300001440770837983734454576162797577879920843837
94800196494427397773038074457387242127660580344021
34682405672126117207785330713414725880593538622654
73975346789773528716600646099932086922357957513492
19756674977305514622634030328067458751013787784885
52829130701073553360162668964463849584071037989913
70486688507994164230666343846674772239240996578903
84568896596451691438428661400890138003041956812810
48591097845251333303104811878603134699136170321975
21568568008502682629819933494624991123862758037223
66019333985981335732480675719265596255491446373799
85178314543906137623414300072437733561943392327043
11884647664463258520202541596231127245136231675622
78619188463016003795639891877182667908801433087053
86290947746644289701355540789696029178210197711709
36545045216872695211681704230649415248953739143673
84812737858390027387981773891609505236508874427301
63038487050158000935975235813062017437144370741479
43915145713587433606973804719686386308885337719433
17041703164332734923888858226543443505628024815340

 

Written by meditationatae

August 26, 2016 at 2:54 am

Posted in History

Mission accomplished in Re: emails (sent…)

copy follows……

 

Subject: Original (carbon copy) of the Notification to Paule Vallieres, Special care counsellor, Quebec City, Canada

 

(original or carbon copy of Notification follows)



NOTICE TO: PAULE VALLIERES, St Brigid’s and Jeffery Hale, Quebec City :



(1) All future meetings in your capacity as Special care counsellor, or
any other capacity, are hereby unilaterally cancelled by me, DAVID
PATRICK BERNIER.



(2) You, Paule Vallieres, are HEREBY instructed by me to never contact
me again.


(3) You, Paule Vallieres, are hereby instructed by me to never attempt
to contact me again.


(4) You, PAULE VALLIERES are hereby instructed by me to never call me
again: I will not answer the phone.


(5) You, PAULE VALLIERES, are hereby instructed by me to never attempt
to call me again.



(6) YOU, PAULE VALLIERES, are hereby instructed to never contact me by
email again.



(7) YOU, PAULE VALLIERES, are hereby instructed to never attempt to
contact me by email again.



(8) YOU, PAULE VALLIERES, are hereby instructed by me to never contact
me by mail again.



(9) YOU, PAULE VALLIERES, are HEREBY instructed by me to never attempt
to contact me by mail again.



(10) YOUR COOPERATION IN THESE MATTERS, PAULE VALLIERES, WILL BE
WELCOMED BY ME.



DONE AT QUEBEC CITY, CANADA, THIS THIRTEENTH DAY OF JULY IN THE YEAR OF
OUR LORD TWO THOUSAND AND SIXTEEN.



AND I HEREBY AFFIX A TOKEN OF MY SIGNATURE:



DAVID PATRICK BERNIER

QUEBEC CITY, PROVINCE OF QUEBEC

CANADA.

===



ORIGINAL/CARBON-COPY BY email TO THE FOLLOWING LIST:



(a) PAULE VALLIERES ***** at ssss.gouv.qc.ca [ Original ]



(b) AIMEE BERGERON:  ***** at ssss.gouv.qc.ca [cc #1 ]



(c) Madame Brigitte Paquette, directrice des services généraux et
communautaires, (DSGC), 1250 chemin Sainte-Foy, Quebec (QC), G1S 2M6

(sous pli separe, by separate cover).  [separate cover #1 ]



(d) my relative A* [ cc #2 ]

(e) my aunt E*    [ cc #3 ]

(f) my late father at jpbern ….  [ cc #4 ]



(g) myself at videotron    [ cc #5 ]


(h) myself at YAHOO mail = ezcos. [ cc #6  ]



SO PROMISED, SO DONE.


“Verba volant, scripta manent.”

Written by meditationatae

July 13, 2016 at 9:39 am

Posted in History

NOTICE TO: PAULE VALLIERES, St Brigid’s and Jeffery Hale, Quebec City

(1) All future meetings in your capacity of Special care counsellor, or any other capacity, are hereby unilaterally cancelled by me, DAVID PATRICK BERNIER.

 

(2) You, Paule Vallieres, are HEREBY instructed by me to never contact me ever again.

(3) You, Paule Vallieres, are hereby instructed by me to never attempt to contact me again.

(4) You, PAULE VALLIERES are hereby instructed by me to never attempt to call me again: I will not answer the phone.

(5) You, PAULE VALLIERES, are hereby instructed by me to never attempt to call me again.

 

(6) YOU, PAULE VALLIERES, are hereby instructed to never contact me by email again.

 

(7) YOU, PAULE VALLIERES, are hereby instructed to never attempt to contact me by email ever again.

 

(8) YOU, PAULE VALLIERES, are hereby instructed by me to never ever contact me by mail again.

 

(9) YOU, PAULE VALLIERES, are HEREBY instructed by me to never ever attempt to contact me by mail again.

 

(10) YOUR COOPERATION IN THESE MATTERS, PAULE VALLIERES, WILL BE WELCOMED BY ME.

 

DONE AT QUEBEC CITY, CANADA, THIS THIRTEENTH DAY OF JULY IN THE YEAR OF OUR LORD TWO THOUSAND AND SIXTEEN.

 

AND I HEREBY AFFIX A TOKEN OF MY SIGNATURE:

DAVID PATRICK BERNIER

QUEBEC CITY, PROVINCE OF QUEBEC

CANADA.

===

 

CARBON-COPY BY email TO THE FOLLOWING LIST:

 

(a) PAULE VALLIERES ***** at ssss.gouv.qc.ca

 

(b) AIMEE BERGERON:  ***** at ssss.gouv.qc.ca

 

(c) Madame Brigitte Paquette, directrice des services généraux et communautaires, (DSGC), 1250 chemin Sainte-Foy, Quebec (QC), G1S 2M6

(sous pli separe, by separate cover).

 

(d) my relative A*

(e) my aunt E*

(f) my late father at jpbern ….

 

(g) myself at videotron

(h) myself at YAHOO mail = ezcos.

 

SO PROMISED, SO DONE.

“Verba volant, scripta manent.”

 

 

Written by meditationatae

July 13, 2016 at 8:23 am

Posted in History

Computing Mertens product in quadruple precision

We can define the Mertens product, Mertens(x), named after Franz Mertens (1840-1927) as:

Mertens(x) := Product_{primes ‘p’ <= x} ( 1 – 1/p) .

A result known as Mertens’ Third Theorem is that , with

C = e^(-gamma), gamma being the Euler-Mascheroni constant, and the number ‘e’ being the base for natural logarithms,

lim_{ x -> oo} Mertens(x)/g(x)  = 1  ,

where

g(x) := C/ln(x) .

For example, Mertens(10^6) is about 0.04063821 and g(10^6) is about 0.04063979.

Srinivasa Ramanujan did his thesis work on highly composite numbers, nominally under the Cambridge mathematician Hardy. This resulted in a manuscript too long to appear as a single paper in 1915 England. So the first part of ” Ramanujan’s Thesis ” was published in 1915, and the second part only in 1997 in a Mathematics Journal.

The 1997 article (posthumous) was annotated by Jean-Louis Nicolas and Guy Robin, two French mathematicians who had both published a areas close Ramanujan’s highly composite numbers (among other things).

The result is the paper:

Highly composite numbers, annotated by Jean-Louis Nicolas and Guy Robin, The Ramanujan Journal volume 1 , number 2, 1997.

For the sections of this second paper dealing with highly composite numbers and colossally abundant numbers, Ramanujan assumers the Riemann Hypothesis. By assuming the Riemann Hypothesis, some delicate estimates can be proven [ by Ramanujan, ].

One of these is formula (362) in section 68, on page 139 of the paper referred to above. Jean-Louis Nicolas answered some of my questions by email on Section 68. Ramanujan writes in a very terse style, it seems.

By doing some routine although very lengthy computations (calculus with special functions, rarely used), I may have improved marginally on Ramanujan’s formula (362).

I wrote about this in the newsgroup sci.math in posts, beginning with:

Approximations to the product in Mertens Third Theorem, and Ramanujan’s “Highly Composite Numbers”, Part 2 (1997)

It’s archived at Google groups,

https://groups.google.com/forum/#!topic/sci.math/YArrppXRlD0

The last post there was dated June 7.

I expect to do numerical computations of Mertens(x) for x up to approximately 10^14, and compare with the approximation of June 7’th that uses the zeta zeros.

The accuracy (or error term) is conditional on the Riemann Hypothesis being true. The Riemann Hypothesis has been verified for the first 10^13 non-trivial zeros.

 

 

Written by meditationatae

July 1, 2016 at 5:48 pm

Posted in History

Quadruple precision computations on asymptotics of colossally abundant numbers

FROM 2015:

 

====

Added July 17, 2015:

In “Highly Composite Numbers” by Ramanujan, annotated by Jean-Louis Nicolas and Guy Robin, The Ramanujan Journal, 1997 one finds the previously unpublished continuation of Ramanujan’s 1915 paper. This continuation can be found in the book:

Ramanujan’s Lost Notebook, Part 3 by George E. Andrews, Bruce C. Berndt

Chapter 10 of this book entitled “Highly Composite Numbers”, essentially contains the 1997 paper in The Ramanujan Journal, annotated by Nicolas and Robin.

The motivation for the graph below can be found in formula 10.71.382 on page 386 of the book by G.E. Andrews and Bruce C. Berndt, as well as in liminf and limsup inequalities immediately following formula (10.71.382).

For colossally abundant numbers ‘N’, Ramanujan finds explicit constants C_1 and C_2 such that, under the assumption that the Riemann Hypothesis holds,

liminf_{ N -> oo, N a C.A. number} -delta(N) sqrt(log(N))  >= C_1 ~= -1.558     and

limsup_{N-> oo, N a C.A. number} -delta(N) sqrt(log(N)) <= C_2 ~= -1.393 .

For 8000 CA numbers N_1, N_2, …, N_j, …  N_8000  we computed:

delta(N_j)  and log(N_j) , 1 <= j <= 8000, where

delta(N_j) = exp(gamma)*log(log(N_j)) – sigma(N_j)/N_j   (as in Briggs, 2006).

The quantities of interest are then the

delta(N_j) sqrt(log(N_j))   for 1<=j <= 8000.

One finds that the mean value M of these 8000 values of delta(N_j) sqrt(log(N_j)) is given by M approximately equal to 1.3881198 .

For 1<= j<= 8000, one defines y_j = delta(N_j) sqrt(log(N_j)) – M and

x_j = log(log(N_j)).  The Graph or plot of the 8000 points (x_j, y_j) , 1<= j <= 8000 is reproduced below.

deltaSquareRootJuly17

====

Added July 18, 2015:

If we look at formula 10.71.382 on page 386 of the book by G.E. Andrews and Bruce C. Berndt, from the continuation of Ramanujan’s (suppressed) 1915 paper “Highly Composite Numbers”, i.e. the annotated paper by Ramanujan from 1997 in The Ramanujan Journal, annotated by Jean-Louis Nicolas and Guy Robin, we see that the fine behaviour of the delta(N) sqrt(log(N))  for colossally abunadant numbers  ‘N’ has a contribution owing to (log(N))^{rho-1}, for rho taking successively the values of the non-trivial zeros of the Riemann zeta function, say ordered by increasing modulus, so as create a real-valued correction, absolutely convergent, and depending on the non-trivial zeros both in the upper half-plane and the lower half-plane.

My method was semi-empirical, in the sense that I chose the sign of the correction term S_1 (log(N)) to best cancel the oscillations in the 8000 terms of

y_j = delta(N_j) sqrt(log(N_j)) – M .

as I recall, this involved adding to delta(N_j) sqrt(log(N_j)) – M

(in PARI/gp notation):

2*exp(Euler)*sum(Y=1,11101,cos(moreImrho[Y]*log(X))/(1/4+moreImrho[Y]^2))

moreImrho[] an array of the imaginary parts of 11101 first non-trivial zeros in upper half-plane, from Andrew Odlyzko’s tables of zeta zeros;

Euler is the Euler-Mascheroni constant 0.577 … ;

We take  X to be log(N_j), for 1<=j<= 8000. This is because in formula 382, Ramanujan applies the S_1 function to log(N), N being a colossally abundant numbers;

More PARI/gp commands:

for(X=1,8000, vz[X] = S5(exp(vx[X])) + vy[X])

the j’th element of the array vx[ ] is actually log(log(N_j)), 1<= j <= 8000;

? \u

S5 =
(X)->2*exp(Euler)*sum(Y=1,11101,cos(moreImrho[Y]*log(X))/(1/4+moreImrho[Y]^2)).

===

In the plot below, the curve in red has as Y values:

y_j = delta(N_j) sqrt(log(N_j)) – M  , M the mean, M = 1.3881198 .

The curve in blue is my attempt at subtracting from delta(N_j) sqrt(log(N_j)) – M the “first order contribution from the non-trivial zeta zeros”, so to say:

deltaSquareRootBBJuly18B

====

Added July 20, 2015:

For the data on 32,000 CA numbers out to about exp(exp(31.09)), I did a plot of

delta(N_j)*sqrt(log(N_j)) – M  for 1<= j <= 32,000 in Red.

The M is the mean from the series for j = 1 to 8000, which was M ~= 1.3881198.

Once again, I made an attempt at subtracting the first order contribution from the zeta zeros.

The resulting rather flat curve is in blue:

xyz32kDraft1

 

========================================================

Added Sunday June 19, 2016:

 

In August 2015, I did computations on colossally abundant numbers out to approximately exp(exp(33)). As above, I substracted the first order contribution of the first 11,101 non-trivial zeta zeros. The resulting blue curve had notable peaks in it for log(log(n)) > 33.

I surmised that this could be due to insufficient precision in the floating point arithmetic, which was done using C “long doubles”, which have a precision of approximately 19 decimal digits.

I therefore decided to redo the computations in quadruple precision, using the math. functions in GNU GCC quadmath library. Even after some optimization such as taking logarithms of products of 256 consecutive primes, the computations of log(N) and sigma(N)/N for 256,000 colossally abundant numbers took approximately 5-6 weeks, without taking into account a re-start of a job that terminated prematurely.

The upshot is that, in quadruple precision, the blue curve has no more suspicious peaks for log(log(n)) > 33, where ‘n’ is a colossally abundant number.

The graphs using quadruple precision arithmetic are shown below.

newcompletequadjune19of2016

Written by meditationatae

June 19, 2016 at 7:37 am

Posted in History

incapsula.com cloud

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Written by meditationatae

June 5, 2016 at 4:37 pm

Posted in History

Firefox messages…

[david2@localhost firefox]$ ./firefox
[aac @ 0x7f6c7af21000] channel element 0.0 is not allocated
[david2@localhost firefox]$ Vector smash protection is enabled.
[NPAPI 28649] ###!!! ABORT: Aborting on channel error.: file /builds/slave/rel-m-esr45-l64_bld-0000000000/build/ipc/glue/MessageChannel.cpp, line 1861
[NPAPI 28649] ###!!! ABORT: Aborting on channel error.: file /builds/slave/rel-m-esr45-l64_bld-0000000000/build/ipc/glue/MessageChannel.cpp, line 1861
Vector smash protection is enabled.
1464974923069 addons.update-checker WARN Update manifest for {972ce4c6-7e08-4474-a285-3208198ce6fd} did not contain an updates property
1464974923579 addons.xpi ERROR Failed to clean updated system add-ons directories.: Unix error 2 during operation DirectoryIterator.prototype.next on file /home/david2/.mozilla/firefox/if6j8gzw.default-1464974038239/features (No such file or directory) ((unknown module)) No traceback available
1464974924095 addons.xpi ERROR Attempted to load bootstrap scope from missing directory /home/david2/.mozilla/firefox/if6j8gzw.default-1464974038239/extensions/firefox-hotfix@mozilla.org.xpi
1464974924096 addons.xpi WARN Add-on firefox-hotfix@mozilla.org is missing bootstrap method shutdown
1464974924098 addons.manager WARN Exception calling callback: [Exception… “Component returned failure code: 0x80520006 (NS_ERROR_FILE_TARGET_DOES_NOT_EXIST) [nsIFile.isDirectory]” nsresult: “0x80520006 (NS_ERROR_FILE_TARGET_DOES_NOT_EXIST)” location: “JS frame :: resource://gre/modules/addons/XPIProvider.jsm :: getURIForResourceInFile :: line 1508” data: no] Stack trace: getURIForResourceInFile()@resource://gre/modules/addons/XPIProvider.jsm:1508 < this.XPIProvider.callBootstrapMethod()@resource://gre/modules/addons/XPIProvider.jsm:4666 < this.XPIProvider.uninstallAddon()@resource://gre/modules/addons/XPIProvider.jsm:4919 < AddonWrapper.prototype.uninstall()@resource://gre/modules/addons/XPIProvider.jsm:7126 < uninstallHotfix/<()@resource://gre/modules/addons/XPIProvider.jsm -> jar:file:///home/david2/.mozilla/firefox/if6j8gzw.default-1464974038239/extensions/firefox-hotfix@mozilla.org.xpi!/bootstrap.js:78 < safeCall()@resource://gre/modules/AddonManager.jsm:179 < makeSafe/<()@resource://gre/modules/AddonManager.jsm:195 < Handler.prototype.process()@resource://gre/modules/Promise.jsm -> resource://gre/modules/Promise-backend.js:933 < this.PromiseWalker.walkerLoop()@resource://gre/modules/Promise.jsm -> resource://gre/modules/Promise-backend.js:812 < this.PromiseWalker.scheduleWalkerLoop/<()@resource://gre/modules/Promise.jsm -> resource://gre/modules/Promise-backend.js:746
1464981075254 addons.xpi WARN Add-on firefox-hotfix@mozilla.org is missing bootstrap method shutdown

Written by meditationatae

June 3, 2016 at 7:14 pm

Posted in History