(a) There certainly do exist pairs of real numbers with and . In fact, it is a result of Ki, Kim and Lee that for , there are infinitely many zeroes of , all but finitely many of which are real.

(b) You have only shown that the equation holds for a single value of , not for all . The evolution of the heat equation does not depend only on the pointwise value of the initial data at a single value of , but on the values at all other positions as well (the heat equation has infinite speed of propagation).

]]>Suppose there exists some pair of real numbers with , such that

$\latex H_{T}(z)=0.$ We shall refer to this as equation . It is a classical fact that as many real zeros, and let be one such zero, where is some real number. That is, We shall refer to this as equation . Combining equations and yields

We shall refer to this as equation As noted in Rodgers and Tao’s paper (page 3), one can view as the evolution of under the backwards heat equation , where denotes the time. Hence from equation we deduce that,“one can view as the EVOLUTION of ” But this quoted statement is meaningless, since both and represent the same time .

We therefore conclude that our supposition must be false, and the desired result follows. $\square$ equal to zero.

]]>For positive t, one has

(see the equation before (35) in https://github.com/km-git-acc/dbn_upper_bound/blob/master/Writeup/debruijn.pdf ).

]]>Since in (1) (for ) can be replaced by , is it possible that this modification of (1) (with the resulting branch point at due to ) still holds for some analytic continuation (with respect to ) of the modified integral ?

]]>Unfortunately, the identity (1) is only valid for (as you point out), hence the identity (2) is only valid when , that is to say it is only established in the case (where it is trivial). Actually, one can check numerically that (2) is false in general.

]]>Indeed, for , we have

Notice that the right-hand side is invariant under the transformation , thus we have

for all . Suppose that for some real hence . But we know by a result of Rodgers and Tao that for any and . Thus we arrive at a contradiction,, which entails that our supposition must be false, and the desired result follows.

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