A Variation Norm Carleson Theorem Along the Primes

Let $\mathrm{\Lambda }$ denote the von Mangoldt function; we prove that for each $r>2$, there exist constants $$r^{\mathrm{\prime }}<\mathbf{c}(r)<2<\mathbf{C}(r),\underset{r\to \mathrm{\infty }}{\lim }\mathbf{c}(r)=1,\underset{r\to \mathrm{\infty }}{\lim }\mathbf{C}(r)=\mathrm{\infty }$$ so that the discrete variational Carleson operator along the primes \begin{align} $\mathcal{V}$^r \Big( \sum_{n \neq 0} f(x-n) Λ(|n|) \frac{e^{2πi λn}}{n} : λ\in $\mathbb{T}$ \Big) \end{align} is bounded on ${\mathrm{\ell }}^p$ for all $\mathbf{c}(r)<p<\mathbf{C}(r)$, while the variation is unbounded when $p\le r^{\mathrm{\prime }}$. At the non-variational endpoint, the same argument gives the sharp maximal result: the prime Carleson operator $$\underset{\lambda \in \mathbb{T}}{\sup }\mid \sum _{n\mathrm{\ne }0}f(x-n)\mathrm{\Lambda }(\mathrm{\mid }n\mathrm{\mid })\frac{e^{2\pi i\lambda n}}{n}\mid$$ is bounded on ${\mathrm{\ell }}^p(\mathbb{Z})$ for the full expected range $1<p<\mathrm{\infty }$. The proof gives a new mechanism for treating modulation-invariant singular integrals after arithmetic sparsification. It combines higher-order Fourier uniformity, a variable-coefficient multi-frequency principle in the spirit of Bourgain, and an additive-combinatorial inverse argument. A key step is a reduction to finite periodic models, where the Ramanujan structure of the major arcs is converted into a sharp estimate for structured atoms by elementary number theory.