Shifted primes in arithmetic progressions to large moduli with linear sieve remainder weights
Ruiyi Zhang · Preprint, 2026-10-24 · 75 pages
PDFHTMLDOI 10.5281/zenodo.23615525Source and checks
Abstract
Let N be a sufficiently large even integer, and fix B > 0 and 0 < ε ≤ 10^(−4). Take Iwaniec's upper or lower linear-sieve remainder weights. We estimate a signed sum of prime-counting discrepancies on N/(log N)^B < p < (1 − ε)N. Fix 0 < u ≤ 1/500 and a sieve exponent 1/2 < θ < (5 − 7/32 − 3u)/8. Choose 0 < σ ≤ min{10^(−10), ((5 − 7/32 − 3u)/8 − θ)/200, sqrt(u/(2θ))}. Then the complete modulus support lies below N^((5−7/32−3u)/8). For every fixed saving exponent A > 0, the error is O(N (log N)^(−A)). For each of the two signs ν ∈ {+, −}, the result holds uniformly for integers a ∈ [N, N log N/3] satisfying (a, q) = 1 for every modulus q in the support of the corresponding coefficient w_N^ν. Call these shifts admissible; they form a subset of the interval that includes a = N. When u = 1/500 and θ = 19101/32000 − 10^(−4), the ratio log D/log z is 298.403125. The proof keeps the smooth-divisor and prime-tuple factors of the linear-sieve weights. It combines dispersion with a Heath–Brown decomposition. The boundary, Type-I and Type-II terms are estimated separately.
Cite
Ruiyi Zhang, Shifted primes in arithmetic progressions to large moduli with linear sieve remainder weights, preprint, Zenodo, 2026. https://doi.org/10.5281/zenodo.23615525