I have to come back to sheldon's ideas when I find the time.
But I wanted to comment quickly that I suspect the truth of a generalized Carlson's theorem.
Carlson talks about f(integer) = 0 but what if we relax this and say the number of zero's on the real line grows like O(a |z| + b) ?
In other words linear ?
Lets call that " strong Carlson ".
Then I assume :
( by lack of knowledge of it being named already )
Tommy-Carlson conjecture :
Let |f(z)| < exp( A |z|^B )
for real A > 0 and some B = 1/m for a positive integer m.
If the number of zero's for f(z) grows like O( p(|z|) ) where p is a real polynomial of degree m then f(z) is Constant.
Now since we know that exp^[0.5](z) < exp( A |z| ^ B ) for sufficiently large A then by combining the previous posts we can conclude the the product expansion for the fake exp^[0.5] is indeed
f(0) ( 1 - z/a_1 ) ( 1 - z/a_2 ) ...
Notice that [1] := exp( A | z^B | ) resembles [2]:= exp ( A |z|^B ).
Because [1] seems to follows from the " Strong Carlson " and the resemblance with [2] I suspect we can say :
" Tommy-Carlson theorem "
In other words the conjecture is probably true !
Even stronger , it is true for EVERY fake exp^[0.5] , not just the one we used here ... but for instance also the one with alternating signs in the derivatives ( see posts 35 , 38 ) !
regards
tommy1729
But I wanted to comment quickly that I suspect the truth of a generalized Carlson's theorem.
Carlson talks about f(integer) = 0 but what if we relax this and say the number of zero's on the real line grows like O(a |z| + b) ?
In other words linear ?
Lets call that " strong Carlson ".
Then I assume :
( by lack of knowledge of it being named already )
Tommy-Carlson conjecture :
Let |f(z)| < exp( A |z|^B )
for real A > 0 and some B = 1/m for a positive integer m.
If the number of zero's for f(z) grows like O( p(|z|) ) where p is a real polynomial of degree m then f(z) is Constant.
Now since we know that exp^[0.5](z) < exp( A |z| ^ B ) for sufficiently large A then by combining the previous posts we can conclude the the product expansion for the fake exp^[0.5] is indeed
f(0) ( 1 - z/a_1 ) ( 1 - z/a_2 ) ...
Notice that [1] := exp( A | z^B | ) resembles [2]:= exp ( A |z|^B ).
Because [1] seems to follows from the " Strong Carlson " and the resemblance with [2] I suspect we can say :
" Tommy-Carlson theorem "
In other words the conjecture is probably true !
Even stronger , it is true for EVERY fake exp^[0.5] , not just the one we used here ... but for instance also the one with alternating signs in the derivatives ( see posts 35 , 38 ) !
regards
tommy1729
