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Let $f\colon\mathbb C\to\mathbb C$ be an entire non constant function. We consider its values on the real line. The order of vanishing (the analytic rank) at a point $s=a$ is denoted by $m$ (the minimal integer $m≥0$ such that $f^{(m)}(a)≠0$)

My question is : Is there exists other statements equivalent to the analytic rank defined in this question?

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This may not be the answer you had in mind, but a very popular subject nevertheless. Just try to google "analytic rank". There is a famous "equivalent formulation" for the analytic rank in a special case: for so-called $L$-functions $L(E,s)$ in the complex variable $s$ of an elliptic curve $E$ over $\mathbb{Q}$ an equivalent statement for the analytic rank $r_a$ of $L(E,s)$ is the algebraic rank $r$, defined by the group isomorphism $E(\mathbb{Q})\simeq \mathbb{Z}^r\oplus T_{tors}$. This is the famous Birch and Swinnerton-Dyer conjecture, i.e., that $r=r_a$.

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