20110306, 00:42  #12  
Dec 2008
you know...around...
2^{5}·19 Posts 
Slightly modified to make it sound more sophisticated :
Quote:
The "appropriate range" should not be too small (<~n^{1/4}) to avoid a ragged graph, but not too big either (>~n/10) to keep n "within the focus". I'll put this into another variable: r(n,p) = q(n,p)q(,p). For , r(n,p) has a  local minimum of about 0.018 at and a  local maximum of about 0.0021 at . Furthermore, for , a  local minimum of about 0.00014 at , preceded by a  local maximum of about 0.00003 at in the range . Enough yet? The preceding minimum should be somewhere around . These are numbers taken from 10^{9} sample values between 10^{16}5*10^{8} and 10^{16}+5*10^{8} and might not be accurate to the last digit if this pattern continues that way for n > . I still can't make head nor tail of this phenomenon, but I would guess these numbers are connected to something which is already known, i.e. can be calculated from wellknown constants. But for now, I've got to get some sleep. 

20200720, 13:37  #13 
Dec 2008
you know...around...
2^{5}×19 Posts 
If there's a single person other than me out there who was stumped by this problem, let me solve it by referring to the Buchstab function. This paper, on page 11, contains the very answers to my question:
https://www.ams.org/journals/mcom/19...99010230438/ (A. Y. Cheer and D. A. Goldston: A differential delay equation arising from the sieve of Eratosthenes)  local minimum at p ~ n^(1/c_{2}) = n^0.3618962566...  local maximum p ~ n^(1/c_{3}) = n^0.288206... and so on and so forth. If it was a message to myself in the past, I'd also suggest looking into the parity problem. P.S.: sorry for the gravedigging, but I had to post it here, I remember that I linked to this thread at least on one occasion. Last fiddled with by mart_r on 20200720 at 13:43 Reason: + link description 
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