On Mon, Mar 24, 2008 at 12:17 PM, wouter meeussen <wouter.meeussen@pandora.be> wrote:
is it just me?
I find it confusing when it is claimed that "all non-trivial zero's of Zeta[x+I y] lie 'on' the line x=1/2 " (* original claim *) since much more seems to be happening. Is that because of my lack of understanding the shorthand used?
I'm having a hard time understanding your shorthand.
I would say three things instead:
1/ only for x=1/2 does the function Zeta[x+ I y] pass through *one
single
intersection point* for each revolution of its modulus [0 , 2Pi]
If you are keeping the function Zeta fixed, and keeping the real part of the argument fixed at 1/2, how are you performing a "revolution of it's modulus"?
And what do you mean by "pass through an intersection point?
As x and y vary, Zeta[x + I y] varies. So Zeta will map paths, either open or closed, in the complex plane, into other paths, open or closed in the complex plane. You seem to be making some claim about the nature of this mapping, but I cannot figure out what the claim is.
2/ and for x=1/2 that special point also 'magically' happens to be
{0,0}
Since I don't know what you mean by the "special point". I can't make any sense of this, either.
3/ apart from x=1/2, Zeta[x+ I y] always 'magically' misses the
point
{0,0} (* equivalent to original claim *)
The claim is that unless x = 1/2, or y = 0 and x is a negataive integer, Zeta[x + I y] is not equal to the complex number 0 (which you can think of as {0,0} if you identify the complex numbers with the real plane). But I don't understand the need to invoke magic. It's as though I said
"The only solutions of x^2 = 9 are 3 and -3",
and you restated this as
"The function f(x+ Iy) = (x+ I y)^2 always 'magically' misses the point (9,0), except when x = +3 or -3 and y = 0"
I suppose it's equivalent, but I don't see what the second formulation adds.
... and, if x>1/2, it crosses y=0 somehere Re[ it ] >0, and if x<1/2,
then
at Re[ it ] <0. Everytime. Ha!
The original formulation leaves it open whether Zeta[1/2+ I y] could ever *not* pass through {0,0} during a complete turn of its modulus (passing somehere through {u, 0} with u < 0.1 say, like between y= 110 and
Ah Andy, you're right. I polar-plot Im( Zeta[1/2+I b] ) on the y axis, and Re( Zeta[1/2+I b] ) on the x-axis, now read me like: only for a=1/2 does the function Zeta[a+ I b] pass through *one single intersection point in the x-y plane* for each revolution of its modulus [0 , 2Pi], but of course plotted as b goes from 12 to 162 and on. see? just need to read what I think ;-)) If you make that plot not at a=1/2, but at a=99/200 and again at a=101/200, then the crossings of y=0 are spread out on the x-axis. Just that. and yes, defining a modulus is trickier than I thought, looking at those short spaced roots (Lehmer's Phenomenon close to b=7005 as Eric Weinstein shows it). The polar plot takes an ugly turn there. it looked nice for the first few tens of non-trivial roots, and my lesson of today is: zeta is even nastier than one thinks. Wouter. ----- Original Message ----- From: "Andy Latto" <andy.latto@pobox.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Monday, March 24, 2008 6:13 PM Subject: Re: [math-fun] Zeta-zoo 112)
Again, you refer to something as "a complete turn of its modulus", by which I think you mean some variation of x, y, or both, (that is, some path in the complex plane), but I cannot tell what path you are referring to.
But, maybe my concept of "revolution around a center from where its modulus
is
seen to monotonously increase" is completely naive. Can such thing be defined for good old Zeta? Would { 0+epsilon, 0 } do?
hmmmppfff (puffing...). Why don't people simply say such things up front?
Wouter.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
__________ NOD32 2968 (20080324) Informatie __________
Dit bericht is gecontroleerd door het NOD32 Antivirus Systeem. http://www.nod32.nl