19-consistent goodness of measurement systems

Here is a list of cons(19, n) for n in the range 1 to 10000. The
function "cons" was calclulated via the following Maple routine:

with(padic,ordp):
cons := proc(w,n)
local i,j,d,e,t,f,h,p,q,u,v;
d := 0;
for i by 2 to w do if isprime(i) then d := d+1; p[d] := i fi od;
e := 0;
for i by 2 to w-2 do
for j from i+2 by 2 to w do
if gcd(i,j) = 1 then e := e+1; q[e] := j/i fi
od
od;
t := ln(2);
for i to d do h[i] := round(evalf(n*ln(p[i])/t)) od;
for i to e do
u[i] := 0; for j to d do u[i] := u[i]+ordp(q[i],p[j])*h[j] od
od;
for i to e do v[i] := abs(evalf(n*ln(q[i])/t-u[i])) od;
f := 0;
for i to e do f := max(f,v[i]) od;
evalf(n^(1/d)*f)
end:

Some idea of the advantages of programming in Maple or similar
computer algebra packages (e.g. Mathematica, Macsyma, Axiom, etc.)
can be discerned from this--the program uses built-in functions to
compute the GCD, determine if an integer is prime, and measure (with
the ordp function) the divisibility of a rational number by a given
prime.

19-consistent measure with cons(19, n) < 1 for n from 1 to 10000

2 .9728631558
7 .9718578174
31 .9504221384
50 .9174777262
80 .9050078510
94 .8876782729
111 .9407450518
121 .8711227952
311 .8268720520
320 .8842773382
364 .9500455011
400 .9876634122
422 .9828907615
436 .9633235237
460 .9202207116
581 .8608767534
742 .9205311588
1178 .8708834628
1578 .8898715197
2000 .8594235879
2460 .9155568082
3395 .9139203767
8539 .8797434553

Nothing on this list really jumps out, though clearly 311 is a very
strong contender even at the 19-limit. Aside from the goodness (or
perhaps it should be badness, since smaller numbers are better)
measure, we might want to note divisibility properties. The 2460
system system is divisible by 12, and 111 and 1578 are at least
divisible by 3, which accomplishes more or less the same thing;
including some systems which have been mentioned but are not on this
list we have:

72 = 6 * 12
111 = 9.25 * 12
612 = 51 * 12
1200 = 100 * 12
1578 = 131.5 * 12
2460 = 205 * 12

If representing other divisions would be useful, we also have:

2460 = 164 * 15 = 60 * 41 = 205 * 12

152 = 8 * 19
171 = 9 * 19
1178 = 62 * 19 = 38 * 31

121 = 5.5 * 22

460 = 10 * 46

742 = 14 * 53

If you count 50 as interesting, you can even add:

80 = 1.6*50, 320 = 6.4*50, 400 = 8*50, 460 = 9.2*50, 1200 = 24*50,
2000 = 40*50, 2460 = 49.2*50, and 3395 = 67.9*50.

Alas, 311 and 8539 are prime!

> Alas, 311 and 8539 are prime!

Gene, does this have anything to do with the zeta function?
I've long wondered why many good ETs are prime... 5, 7, 19,
31, 41, 53... notable exceptions being 12, 22, 34, 58, and 72...

I know Paul's been asking for your zeta stuff; did I miss it?

-Carl

--- In tuning-math@y..., "Carl" <carl@l...> wrote:

> Gene, does this have anything to do with the zeta function?

Only in the sense that the big spikes in the zeta function occur at
good et's. The zeta function doesn't have the mojo to do this any
better than we can do in a much less abstruse way in any case.

> I know Paul's been asking for your zeta stuff; did I miss it?

I'd like to find a good program for computing zeta(s+it) and Z(t)
(which you might say is zeta(1/2+it) made into a real analytic
function of a real variable.) I don't want to go back and write my
own again!

I think they may be availble for Mathematica, but I'd like to find a
Maple version.

--- In tuning-math@y..., genewardsmith@j... wrote:

> I'd like to find a good program for computing zeta(s+it) and Z(t)
> (which you might say is zeta(1/2+it) made into a real analytic
> function of a real variable.) I don't want to go back and write my
> own again!

I just tested the Maple zeta function routine, and it *does* seem to
work for complex arguments--I had thought otherwise. I'm off to look
at some graphs of et's, and will report anon.

>> Gene, does this have anything to do with the zeta function?
>
> Only in the sense that the big spikes in the zeta function occur
> at good et's.

That's by definition, right? (that is, doesn't actually explain
anything).

-Carl

>I just tested the Maple zeta function routine, and it *does* seem
>to work for complex arguments--I had thought otherwise. I'm off to
>look at some graphs of et's, and will report anon.

Cool!
-C.