Fixed-shift Sathe–Selberg asymptotic for integers

Establish, for every fixed sequence of k distinct non-negative integers h₁,…,h_k and every real number R with 0<R<2, the uniform asymptotic formula ∑_{n≤N} z^{Ω(n+h₁)+⋯+Ω(n+h_k)} = G(z)^k N(log N)^{k(z−1)} + O(N(log N)^{k(Re z−1)−1}) for |z|≤R and N≥2.

Background

The paper proves an averaged Sathe–Selberg-type formula after summing over the shifts h₁,…,h_k. It then identifies the corresponding single-sum formula for any fixed distinct shifts as an unresolved conjectural strengthening. The paper states that this estimate would imply the pointwise dynamical generalization of Chowla’s conjecture formulated earlier.

References

By \cref{Selberg1954_Chowla}, we conjecture that that uniformly for $|z|\leq R$ and $N\ge2$, we have \begin{equation}\label{Sathe_Selberg_Chowla} \sum_{n\leq N} z{\Omega(n+h_1)+\cdots + \Omega(n+h_k)} = G(z)k N(\log N){k(z-1)} + O\big(N (\log N){k(Re z-1)-1} \big). \end{equation}

Sathe_Selberg_Chowla:

∑n≤NzΩ(n+h1)+⋯+Ω(n+hk)=G(z)kN(log⁡N)k(z−1)+O(N(log⁡N)k(Rez−1)−1).\sum_{n\leq N} z^{\Omega(n+h_1)+\cdots + \Omega(n+h_k)} = G(z)^k N(\log N)^{k(z-1)} + O\big(N (\log N)^{k(Re z-1)-1} \big).

— A dynamical generalization of Chowla's conjecture on average  (2608.16108 - Wang, 17 Aug 2026) in Remark following Theorem 2.2, Section 2, “The Sathe-Selberg type theorems”

Similarly, by \cref{Selberg1954_Chowla_prime}, we conjecture that that uniformly for $|z|\leq R$ and $N\ge2$, we have \begin{equation}\label{Sathe_Selberg_Chowla_prime} \sum_{p\leq N} z{\Omega(p+h_1)+\cdots + \Omega(p+h_k)} = G(z)k N(\log N){k(z-1)-1} + O\big(N (\log N){k(Re z-1)-2} \big). \end{equation}

Sathe_Selberg_Chowla_prime:

∑p≤NzΩ(p+h1)+⋯+Ω(p+hk)=G(z)kN(log⁡N)k(z−1)−1+O(N(log⁡N)k(Rez−1)−2).\sum_{p\leq N} z^{\Omega(p+h_1)+\cdots + \Omega(p+h_k)} = G(z)^k N(\log N)^{k(z-1)-1} + O\big(N (\log N)^{k(Re z-1)-2} \big).

— A dynamical generalization of Chowla's conjecture on average  (2608.16108 - Wang, 17 Aug 2026) in Remark following Theorem 2.3, Section 2, “The Sathe-Selberg type theorems”