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