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.