Dear Enrique,

yes it has been published.

《Journal of Combinatorics & Number Theory》, 2010, 3(1):33-38

7-free has been published as well. Some australian did 11-free.

Best Regards

Patrick

Thank you!  I am preparing a document with my coauthor for Prof. Nicolas which I might as well share with both of you.  It provides strong numerical evidence to the correctness of the Dedekind criterion (if we give it a name).  Furthermore, I have identified a function which is a lower bound to the LHS, but this needs to be proved.  It is an easy function...but how to prove it...more later when all this work is done.

Sincerely,

Enrique

U. of Helsinki, Finland

Dear Patrick,

Could you send me the reference to the 7-free and the 11-free published by an Australian.  I thank  you greatly in advance.

I am still working on the document that provides evidence for the Dedekind criterion.  It is very convincing computer based evidence.

Thanks once again,

Best regards - Enrique

you wanna get article A65 in

http://www.integers-ejcnt.org/vol11.html

and article  A12 in

http://www.integers-ejcnt.org/vol15.html

Best Wishes

Patrick

Thank you very much indeed

Regards,

Enrique

Hello Patrick,

In your seven-free article, your theorem 12, could t=o(log log n) be replaced by t~log log n?  It seems to hold, numerically at least.  And, if you assume the RH, would there be a predicted value of the ratio t/log log n?  I thought I would ask you first, before doing too much that might result in a wild goose chase, since you have worked closely on this.

Thank you for your answer, if it is possible.

Sincerely,

Enrique

Hi Enrique, is your Conjecture

Thm: Almost all n are clog log n -free for some constant c>0.

?

Well, if you phrase it that way, numerical experiments would lead me to think that if, for example, c=2, this might be the case, in other words, for CA n, with n>N, some fixed N, n is 2 log log n - free... does this make sense?  However, I don't see how to prove this.  The problem is that I can't tell, given lack of knowledge of the distribution of primes all the way up, whether this is going to hold.  And then again, I don't know how that will affect your theorem, ie., whether for such a family of numbers the Robin inequality will hold...but consider me a beginner in these matters, I just recently started reading your papers (including the other one about the Dedekind criterion) and think about it now and then, when I have time.  What I have done is put it to the test on a computer up to very high numbers.  Both, the above, and the inequality you express involving the Dedekind function in another paper (Extreme values of the Dedekind function) seem to hold, and seem to hold quite well (in particular, that in the other paper...but about that another time).

Thanks -- E.

Dear Enrique,

of course Jean-Louis Nicolas is the best expert in these questions. If you read his papers on CA numbers maybe enough is known on their prime factorization to show this.

Best Wishes

Patrick

PS You might enjoy my recent work on Asymptotic Robin

https://arxiv.org/pdf/1602.03384.pdf

Dear Patrick,

Thank you!  I started reading it.  

In the meantime, I should tell you, I think I have a proof of CA n being 2 log log n free (which I get with the aid of a computer CAS).  I did it using the approximations that appear in one of Robin's papers, and one of Nicolas's papers (with Sondow and Caveney).  Although there is an inequality along the way that I can't prove, I know that the functions are smooth and increasing, and I can get the limit of their ratio, which is a nice number (log 4).  Implying that the inequality should hold throughout the positive integer domain.

The question I now have in the back of my mind is, would this imply that the CA numbers obey the Robin inequality?  Do you know the answer to this?  I haven't really sat down and read the details of your theorem 12.

I can send you a brief write up, for you to check and perhaps complete? 

Best regards

Enrique

Please send me your write up! I will be happy to read it.

[email protected]

Actually if RH is true by an old result of Ramanujan we know Robin inequality holds. See Section 4, p. 5 of my arxiv paper.

Ok, let me put it together.  This weekend I am busy, but I should have some time to get it done next week.  It will not be in tex, but rather, text from my CAS.  I will see if it can be exported as a tex file.  It will be brief, meaning, some of the details of the functions that I use I only reference to the papers from which they appear.  I will send it to the address you sent above.

Best regards -- Enrique

Right, that we know.  The question was, however, whether your theorem 12 could be modified so that the CA  numbers n obey the Robin inequality (without assuming RH) due to the fact that they are c log log n - free, (I think c needs to be at most 2/log 2, which is the case if c=2).  So I guess that is the next step in this direction.  I need to read and understand your theorem 12 carefully, and see if it can be modified somehow.  But perhaps you know already, in which case, we have an interesting situation - to be carefully analyzed before anything.

Best -- Enrique

Hello, Patrick,

Well, I am trying to write things here, but I note that in your theorem 12 of the 7-free paper you require that the sequence Sn>Nn, where Nn is the product of the first n primes.  Clearly, the sequence of colosally abundant numbers does not have this property, it will have to be that Sn

Hello Patrick,

I must apologize by some confusion I might have caused, and comment on your proposed conjecture (that perhaps tt/log p_n> 1.  In fact, I can work theoretically with the function F defined in Caveney, Nicolas, and Sondow's paper, and get the limit of t/log p_n for CA numbers which have p_n as largest prime factor as n goes to infinity, and that would be 1/log 2.  Likewise,  2 log p_n/t > 1, and the limit of the ratio 2 log p_n/t would be log 4.  This is much weaker than what you propose, and in fact it almost shows that you cannot get t to grow so slowly as to apply your theorem 12 in any way (besides the fact that the sequence of CA numbers grows more slowly than that of the primorials), which is what you actually say might be the case at the end of your paper.  The question remains whether this information in any way helps establish whether the CA numbers obey the Robin inequality.   I am working on that.  But I do not have an answer as yet, maybe never will.  The interesting thing to observe is that p_n ~ log N_n, the n-th primorial (or as it is given in a different notation, p_n#) as you rightly observe in your theorem 12, so in a sense, what we have is t ~ log log N_n, rather than t ~ log log n... and that is the story of how I got confused ... 

So again, this is what I do have, and I apologize for the confusion.  I hope that in looking back a second time I don't find out that I am even more confused than I was there for a moment, in a spark of enthusiasm the conjecture, or let us call it speculation, which I thought might be true.

As to the data regarding the "Extreme values of the Dedekind psi function...", that does hold very nicely, incredibly nicely.  I could almost give you a function that lower bounds the resulting expression if you move everything in the inequality to the left hand side to have an inequality where you have "expression > 0".  I have, however, no clue how to prove it.  Again, the problem here is that one would seem to require prior knowledge of how the primes behave, and that is what we are trying to prove.  It is a bit as if we have a circular problem...However, if a picture were worth a proof ... 

Best regards, and many thanks - Enrique

Dear Enrique,

you seem to be working very hard. I hope you can make a publication of your numerical results. Do you know the paper by Briggs in Experimental Math on Robin inequality? Are his methods inspiring to you?

Best Regards

Patrick

Dear Patrick,

I am familiar with the paper, but there are some things about it that intrigue me.  For example, he claims to compute CA numbers up to 10^10^10, but not higher.  The log log of that is between 23 and 24.  However, his figure 3, where he discusses delta, has log log n (where n is a CA number) going up to 27, which would bring n closer to 10^10^12 (a "bit" less).  There are other questions I would have.  What I like the best in it though is the idea on how to compactly represent a CA number.  

Regards -- Enrique

Dear Patrick,

I wonder if there is a published version of your joint paper with Planat on extreme values of the Dedekind function. I would like to cite some of your results in it in a paper that I wish to publish. I thank you in advance for your help.

Best regards,

Enrique

Dear Patrick,

Please ignore my previous question. I realize that was the first question I asked in this conversation and that you have already answered it in the affirmative. Thank you, and I apologize for the inconvenience.

Best regards,

Enrique

Dear Patrick,

I meant to ask about the paper "On Nicolas criterion for Riemann Hypothesis", is that published in a journal. Apologies for the confusion. Again, I thank you in advance.

Best regards,

Enrique

"On Nicolas criterion for Riemann Hypothesis", is not published in any journal. Somebody pointed out that some paper of jean-Louis Nicolas solves some of the questions there; but my memory is very vague sorry.