17 Mollified Dirichlet Polynomial
The mollified Dirichlet polynomial \(S_w(T)\) provides a computable approximation to \(S(T) = \pi ^{-1}\arg \zeta (\frac{1}{2} + iT)\) using a finite sum over primes. This chapter formalizes the constructive core (Layer B): the mollifier kernel, the finite sum, and the adaptive cutoff. All results in this section are fully proven in Lean 4 with zero axioms.
17.1 Cosine-Squared Mollifier Kernel
The cosine-squared mollifier kernel:
\(w(x) \geq 0\) for all \(x \in \mathbb {R}\).
\(w(x) \leq 1\) for all \(x \in \mathbb {R}\).
\(w(0) = 1\).
\(w(x) = 0\) for all \(x \geq 1\).
17.2 Mollified Dirichlet Polynomial
For prime \(p\) and power \(m\):
\(S_w(T; \theta )\) is a finite sum and therefore always a well-defined real number. Unlike the formal Dirichlet series \(-\zeta '/\zeta (s)\) which diverges on \(\mathrm{Re}(s) = \frac{1}{2}\), no convergence issues arise.
The \(R^2\) contribution by prime power order \(m\) is:
|
\(m\) |
\(\Delta R^2\) |
Fraction |
|
1 (primes) |
0.872 |
92.8% |
|
2 (squares) |
0.057 |
6.1% |
|
3 (cubes) |
0.011 |
1.1% |
This justifies the standard choice \(K = 3\).
17.3 Adaptive Cutoff
The adaptive cutoff exponent:
determined by the normalization constraint \(\alpha = 1\) uniformly across height windows.
\(S_w(T) = S_w(T; \theta (T))\) using the adaptive cutoff.
Three related cutoff models appear across the paper:
|
Model |
Value |
Context |
|
Constant \(\theta ^*\) (mpmath) |
0.9941 |
§3.5 (100K zeros) |
|
Constant \(\theta ^*\) (Odlyzko) |
0.9640 |
§9 (2M zeros) |
|
Adaptive \((\theta _0, \theta _1)\) |
\((1.409, -3.954)\) |
§4 (recommended) |
17.4 Main Results (Numerical Certificates)
Over the first 100,000 zeros: \(\max _n |N_{\text{approx}}(T_n) - n| = 0.111 {\lt} 0.5\) (safety factor \(4.52\times \)).
98.0% of zeros are localized to their correct inter-Gram interval. The 2.0% failure rate is predicted (within 8%) by a simplified GUE model.
The adaptive formula explains 93.9% of the variance in zero corrections \(\delta _n\) (\(R^2 = 0.939\), out-of-sample \(R^2 = 0.919\)).
Complete verification of the constructive core: kernel properties + sum well-definedness + standard \(K = 3\).