## 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.”

## 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.”

## 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.

## 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:

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:

====

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:

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

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.

## incapsula.com cloud

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## 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

## On counts of finite sequences with discrepancy <= 2

In sci.math, Richard Tobin provided the number of essentially different sequences of discrepancy at most 2, for lengths n from 1 to 75 inclusively.

This leads to a function count(n) , for n = 1 … 75, which is the number of discrepancy D <= 2 sequences of length ‘n’, up to a change of sign.

The growth is very approximately exponential.

Therefore, it’s suitable to define:

f(n) := log( count(n) ).

For example, count(1) = 1, and f(1) = 0.

We now plot the growth, i.e. f(n), minus (the average growth at ‘n’).

This leads to defining:

g(n):= f(n) – C*(n-1) ,

where C is the unique real number such that:

f(75) – C*(75-1) = 0.

Note that we also have g(1) = 0.