Determine the sharp constant in the two-parameter totient sum

Prove a lower bound of the form W(J)\ge(0.45+o(1))(\log J)^2 for W(J)=\sum_{u,a\le J,\,\gcd(u,a)=1}1/\varphi(4ua), thereby improving the constants in the large-sieve exceptional-set estimates.

Background

The paper uses the elementary inequality \varphi(m)\le m to obtain W(J)\ge(\kappa_1+o(1))(\log J)2 with \kappa_1=3/(2\pi2)\approx0.152. Numerical computations suggest that the true limiting constant for W(J)/(\log J)2 is substantially larger, near 0.46.

Establishing the stronger lower bound would improve the constants in the growing-depth large-sieve results, including the exponent in the optimized estimate from approximately 0.065 to approximately 0.093. The authors indicate that a two-variable extension of the method used for the one-variable sum in Lemma 5.1 should suffice, and that no matching upper bound is required.

References

Four questions are left open by the analysis above, in decreasing order of tractability. The constant in Lemma~\ref{lem:kappa1.} The bound $W(J)\ge(\kappa_1+o(1))(\log J){2}$ with $\kappa_1=3/(2\pi{2})$ comes from the crude estimate $\varphi(m)\le m$; the true limit of $W(J)/(\log J){2}$ appears numerically to be close to $0.46$ (Section~\ref{sec:7}, item (8)). Proving $W(J)\ge(0.45+o(1))(\log J){2}$, a finite computation, by the method of Lemma~\ref{lem:kappa0} extended to two variables, would improve every constant of Section~\ref{sec:5.2}, taking the exponent of Corollary~\ref{cor:optimal_tradeoff} from $0.065$ to about $0.093$.

— Sieve dimension and search depth for the Erdős-Straus conjecture, $n \equiv 1 \pmod{24}$  (2608.24035 - Dahan, 25 Aug 2026) in Section 8, “Conclusion and open questions”; Lemma 5.1 and Section 7, item (8)