Prime-sum growth and elliptic-curve rank

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $E/\mathbb Q$ be an elliptic curve, represented by an integral equation $y^2=x^3+Ax+B$ with $A,B\in\mathbb Z$ and $4A^3+27B^2\ne0$. For every prime $p$, define \[ N_p=1+\#\{(x,y)\in\mathbb F_p^2:y^2=x^3+Ax+B\},\qquad f_E(X)=\sum_{\substack{p\text{ prime}\\p\le X}}\log\!\left(\frac{N_p}{p}\right)\quad(X\ge2). \] The extra one accounts for the point at infinity: at good primes $N_p=\#E(\mathbb F_p)$, rather than this projective count plus one. At the finitely many bad primes the same displayed affine-count formula specifies $N_p$. A change to another integral model affects at most finitely many local counts and therefore changes $f_E(X)$ by a constant for all sufficiently large $X$. Bounded means that $|f_E(X)|\le C_E$ for all real $X\ge2$, for some finite constant $C_E$.

Let $r=\operatorname{rank}E(\mathbb Q)$, so that $E(\mathbb Q)$ is the direct sum of a finite torsion group and $r$ copies of $\mathbb Z$.

Statement recorded in the supplied source.

For every elliptic curve $E/\mathbb Q$ of rank $r$, does the logarithmic growth law \[ \lim_{X\to\infty}\frac{f_E(X)}{\log\log X}=r \] hold? For $r>0$ this says $f_E(X)\sim r\log\log X$; at $r=0$ the limit formulation makes the growth coefficient meaningful. [S2]

Short English statement

Is the rank of an elliptic curve the coefficient of double-logarithmic growth of this prime sum?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8900014 (AMR-088-0014). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Christian Wuthrich, Open problems in number theory, lecture slides (December 2011) final slides headed Birch and Swinnerton-Dyer conjecture: $f(X)$ grows like $r\log(\log X)$. https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf.