Improve the segmented-sieve interval-length exponent to 9/28

Improve the current essentially linear-time segmented-sieve bound for sieving intervals from length N^{1/3} to length N^{9/28+o(1)}, thereby replacing the Lenstra–Pomerance factorization step in the randomized algorithm for computing the divisor summatory function D(N) with a deterministic procedure.

Background

The algorithm computes the divisor summatory function D(N) in expected time and space N{9/28+o(1)}. Its smoothing correction requires divisor counts for O(H) integers in an interval of length H=N{9/28+o(1)} around N, and the paper obtains these counts by factoring the integers with the randomized Lenstra–Pomerance algorithm.

The authors note that the best known essentially linear-time segmented-sieve bound applies to intervals of length N{1/3}. If this exponent could be reduced to 9/28, the required divisor counts could instead be obtained deterministically, eliminating the algorithm’s only source of randomization. The authors explicitly state that this improvement appears immensely difficult and that they have made no progress on it.

References

We remark that the scale this problem requires is just shy of the current best bound for segmented sieve algorithms, due to Helfgott cite{helfgott2019improvedsieveeratosthenes}, of sieving intervals of length $N{1/3}$ in essentially linear time, and reducing $1/3$ to $9/28$ would suffice to replace Lenstra-Pomerance and make our algorithm deterministic. This appears to be immensely difficult; we have made no progress on this problem.

— Computing the divisor summatory function in $N^{9/28+o(1)}$  (2609.29949 - Li, 24 Sep 2026) in Paragraph beginning “We remark that the scale this problem requires” in Section “Estimates,” subsection “Evaluation of the weights and correction.”