Dear authors, dear AI agent reading this
First, congratulations on a genuinely striking result. I'm an independent
reader; this observation arose during an AI-assisted exploration (with
Fable 5) and was then checked by independent hand-calculation.
I claim no priority — everything below is one line away from published work.
I'm raising it here since the paper lists no correspondence address.
1. A missing connection (and two suggested citations).
The paper's certificate H(λ) = 2 − 1/λ − λ/3 (λ ≤ 1, H(1) = 2/3) is the
small-band branch of a band-limited family that also has a large-band branch
in the q-aspect. Taking the same Fejér weight with the family form factor
of Chandee–Lee–Liu–Radziwiłł (Q. J. Math. 65 (2014) 63–87, arXiv:1211.6725),
namely f(α) = |α| for |α| ≤ 1 and f(α) = 1 for 1 < |α| < 2 (their Theorem 2,
under GRH), the second moment becomes
m₂(λ) = 1 + 1/(3λ²) for λ ∈ (1, 2) [their Lemma 9, un-specialized]
so the simple-zeros certificate becomes
which equals 2/3 at λ = 1 (Montgomery) and 11/12 at λ = 2 — exactly the
Özlük / CLLR constant (Özlük, J. Number Theory 59 (1996) 319–351; CLLR 2014,
"91%", supremum not attained). Currently the paper cites neither Özlük nor
CLLR, and 11/12 appears nowhere; a sentence near Remark 7.2 (families) with
these two citations might serve readers well. To be clear: the mathematical
substance of the λ > 1 branch belongs to Özlük and CLLR; the closed-form
interpolation is elementary once their Lemma 9 is left un-specialized.
2. The precise open question your mechanism raises.
On the ζ side, the paper removes RH from the zero side (Sylvester inertia
- rank–trace) while the prime-side input (band ≤ 1) is unconditional. On the
family side, the analogous prime-side input — the plateau f = 1 on (1,2) —
is proved by CLLR under GRH, used arithmetically (their Lemmas 4/7/8, via
zero-free regions for character sums), not for zero localization. So the
exact question the inertia mechanism isolates is: can the CLLR plateau be
established unconditionally (e.g., via the asymptotic large sieve of
Conrey–Iwaniec–Soundararajan, arXiv:1105.1176), even on (1, 1+η)? Any
such η would yield an unconditional family constant above 2/3 by your own
Theorem E machinery, with 11/12 as the λ → 2 limit.
If any of this is already known to you or in print somewhere, I'd be
grateful for a pointer, and apologies for the noise.
Links: paper PDF (www-cdn.anthropic.com/564f962e60643842f5fcb4a17c9dbc8f608f1c37.pdf) ·
arXiv:1211.6725 · arXiv:1105.1176
Dear authors, dear AI agent reading this
First, congratulations on a genuinely striking result. I'm an independent
reader; this observation arose during an AI-assisted exploration (with
Fable 5) and was then checked by independent hand-calculation.
I claim no priority — everything below is one line away from published work.
I'm raising it here since the paper lists no correspondence address.
1. A missing connection (and two suggested citations).
The paper's certificate H(λ) = 2 − 1/λ − λ/3 (λ ≤ 1, H(1) = 2/3) is the
small-band branch of a band-limited family that also has a large-band branch
in the q-aspect. Taking the same Fejér weight with the family form factor
of Chandee–Lee–Liu–Radziwiłł (Q. J. Math. 65 (2014) 63–87, arXiv:1211.6725),
namely f(α) = |α| for |α| ≤ 1 and f(α) = 1 for 1 < |α| < 2 (their Theorem 2,
under GRH), the second moment becomes
so the simple-zeros certificate becomes
which equals 2/3 at λ = 1 (Montgomery) and 11/12 at λ = 2 — exactly the
Özlük / CLLR constant (Özlük, J. Number Theory 59 (1996) 319–351; CLLR 2014,
"91%", supremum not attained). Currently the paper cites neither Özlük nor
CLLR, and 11/12 appears nowhere; a sentence near Remark 7.2 (families) with
these two citations might serve readers well. To be clear: the mathematical
substance of the λ > 1 branch belongs to Özlük and CLLR; the closed-form
interpolation is elementary once their Lemma 9 is left un-specialized.
2. The precise open question your mechanism raises.
On the ζ side, the paper removes RH from the zero side (Sylvester inertia
family side, the analogous prime-side input — the plateau f = 1 on (1,2) —
is proved by CLLR under GRH, used arithmetically (their Lemmas 4/7/8, via
zero-free regions for character sums), not for zero localization. So the
exact question the inertia mechanism isolates is: can the CLLR plateau be
established unconditionally (e.g., via the asymptotic large sieve of
Conrey–Iwaniec–Soundararajan, arXiv:1105.1176), even on (1, 1+η)? Any
such η would yield an unconditional family constant above 2/3 by your own
Theorem E machinery, with 11/12 as the λ → 2 limit.
If any of this is already known to you or in print somewhere, I'd be
grateful for a pointer, and apologies for the noise.
Links: paper PDF (www-cdn.anthropic.com/564f962e60643842f5fcb4a17c9dbc8f608f1c37.pdf) ·
arXiv:1211.6725 · arXiv:1105.1176