Chen 1973, Lemma 6, equations (14) and (15): scalar payments #
This file pays the elementary scalar estimates left after the exact finite
expansions in Chen1973Lemma6Equations14And15. Each payment is a separate
result. The coefficient estimates apply only to Chen's literal coefficients;
there is no arbitrary-coefficient strengthening.
Trivial harmonic majorant for Chen's literal Möbius polynomial on the
Re(s) ≥ 1 line used in equation (14).
Pointwise Abel--Pólya--Vinogradov payment for the literal remainder in
(14). The source range Re(s) ≥ 1 is explicit.
Logarithmic form of the Abel--PV remainder payment. Positivity of H
is stated because it is exactly what makes 1 + log H a nonnegative majorant
for the finite harmonic factor.
The finite convolution coefficient C_H(m)/mˢ in (14) has the same
weighted divisor majorant as the source uses. The hypothesis Re(s) ≥ 1 is
the literal equation-(14) range.
The C_H divisor-energy payment in equation (14).
The literal coefficient j(m) in (15), including its m⁻ˢ weight, is
bounded by τ(m)m⁻¹/² on Chen's half-plane.
Weighted divisor-square payment for the literal j(m) coefficients in
(15). This is the finite form of |j(m)| ≤ τ(m) followed by
∑ τ(m)²/m ≪ log⁴.
Explicit logarithmic version of the (15) coefficient energy.
Equation (15) after paying its literal weighted divisor-square energy.
The finite Abel--PV remainder summed over Chen's literal conductor interval
D < d ≤ Q. The hypotheses 0 < H, 0 < D, and D < Q are exactly the
nonempty source regime; the constant 40 and every height factor remain
visible.
Final scalar form of equation (14). It combines the sharp finite second
moment, the literal C_H divisor energy, and the summed Abel--PV remainder.
No equation-(14)-shaped hypothesis is retained.
Final log⁴ endpoint for equation (15), with Chen's actual half-plane
encoded by Chen1973Lemma3Domain and the literal H,D,Q ranges unchanged.