  Motivated by the recent proof of Newman's conjecture \cite{R-T} we study
certain properties of entire caloric functions, namely solutions of the heat
equation $\partial_t F = \partial_z^2 F$ which are entire in $z$ and $t$. As a
prerequisite, we establish some general properties of the order and type of an
entire function. Then, we start our inquiry on entire caloric functions by
determining the necessary and sufficient condition for a function $f(z)$ to be
the initial condition of an entire solutions of the heat equation and,
subsequently, we examine the relation of the $z$-order and $z$-type of an
entire caloric function $F(t, z)$, viewed as function of $z$, to its $t$-order
and $t$-type respectively, if it is viewed as function of $t$. After that, we
shift our attention to the zeros $z_k(t)$ of an entire caloric function $F(t,
z)$, viewed as function of $z$. We show that the points $(t, z)$ at which $F(t,
z) = \partial_z F(t, z) = 0$ form a discrete set in $\mathbb{C}^2$ and we
derive the $t$-evolution equations of the zeros of $F(t, z)$. These are
differential equations which hold for all but countably many $t \in
\mathbb{C}$.
