1 4th July 11:01 jean-claude evard External User   Posts: 1 Minima of Phi(m) over m Is it already known that over an interval of integers [2, N], the ratio Phi(m) over m is minimum when m is the largest primorial that lies in this interval ? Here: Phi(m) = Eulers function = Number of integers k such that 0

 2 4th July 11:02 gottfried helms External User   Posts: 1 Minima of Phi(m) over m Am 29.06.05 16:30 schrieb Jean-Claude Evard: Without being able to answer the question at this time, is this also true, if the primorial involves the wieferich prime 1093 n1= 2*3*5*...*1093 and the interval [2..N] contains also the number n2=n1*1093? Gottfried Helms
 3 4th July 11:02 jean-claude evard External User   Posts: 1 Minima of Phi(m) over m ----------------------------------------------------- Your question is quite a test for my statement, and I thank you a lot. I do not have a valid proof of my statement yet, but I need the result for my current research work. I wanted first to be sure that it was not already published. I should have called it conjecture. My updated conjectures are the following: Let r(m) = [The ratio phi(m) over m]. Conjecture 1. The relative minimum values of r(m) occur at the points where m is a primorial, and the sequence of values of r(m) at these points is strictly decreasing. Conjecture 2. The relative maximum values of r(m) occur at the points where m is a prime, the sequence of values of r(m) at these points is strictly increasing, and its limit when m goes to infinity is 1. Conjecture 3. If we eliminate the points where r(m) has a relative maximum value, then the new relative maximum values of r(m) occur at the points where m is the square of a prime, the sequence of values of r(m) at these points is strictly increasing, and its limit when m goes to infinity is 1. Everyone is welcome to publish a proof or a counterexample of any of these conjectures. I will try later if it is not already done. I remain interested in any related references. With many thanks for your time and attention, Jean-Claude Evard Department of Mathematics Western Kentucky University
 4 26th July 01:42 jean-claude evard External User   Posts: 1 Minima of Phi(m) over m Below is a copy of an answer to my posting about phi(m)/m from Gerd Verbouwe (Belgium). For technical reasons, he could not get it posted here, and he e-mailed it to me the day of my posting, on June 30, 2005. After a long delay due to emergencies, I checked his answer, and with his agreement, I submit it for posting here. Jean-Claude Evard Department of Mathematics Western Kentucky University ----------------------------------------------------- First, I recall the notation that I used: ----------------------------------------------------- Phi(m) = [The Eulers function] = [The number of integers k such that 0 p occurs, it can be replaced by a smaller one, hence no minimum in this case. [c] Hence (*) is a relative minimum, when as much factors (1-1/prime) as possible occur, and since 1-1/P > 1-1/p for P>p, one needs the first N primesâ€¦ . Of course these are "local" minima, since, e.g. r(2* 17#) = r(17#) and (*) shows immediately that it is strictly decreasing. (Also, becomes clear: it's maximal when only one factor (1-1/p) appears, this happens first for the prime p itself; the limit of these is of course 1.) Am I missing something here? gv ----------------------------------------------------------------------- I think that this proof is 100% OK. I have not checked my third conjecture yet, and I intend to come back to all this during next summer. Jean-Claude Evard