I hope everyone can offer some suggestions for the paper and also bring new ideas.
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刘公善
Nov 17, 2025, 10:20:53 AMNov 17
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We establish a universal state equation governing the spacing of Lehmer pairs—exceptionally
close consecutive zeros of the Riemann zeta function—revealing a hierarchical structure connect-
ing phase disorder, prime pair correlations, and field-theoretic principles. Through systematic
analysis of the first 10,000 non-trivial zeros, we discover that spacing obeys a thermodynamic-like
relation at three progressive levels.
Level I (Base Equation): For Lehmer pairs in the platform region defined by prime phase
alignment, normalized spacing ∆γ follows ∆γ = ∆γ0 + κ · I, where I is a disorder parameter
constructed from phases ϕp(γ) = [γ log p/2π] mod 1 at the optimal prime set {5, 7, 11}. This
achieves R2 = 0.225 with p < 10−20 on n = 336 platform pairs.
Level II (Prime Pairs): Extending the model to include twin (k = 2), cousin (k = 4),
and sexy (k = 6) prime pair coherence terms dramatically improves predictive power to R2 =
0.335—a 56% increase. Twin primes alone contribute ∆R2 = +0.10 (46% improvement, p <
10−11). All three pair types exhibit negative coefficients, indicating that phase coherence reduces
spacing, establishing a quantitative link between the Hardy-Littlewood k-tuple conjecture and
zero distribution.
Level III (Field Theory): The hierarchical structure suggests an analogy to general
relativity, where the linear equation represents a “Newtonian limit” of a deeper field equation.
We observe scale dependence of the coupling constant κ (varying 21% across γ ranges) and
identify a mathematical cosmological constant Λ(math) ≈ π, hinting at fundamental connections
to quantum field theory’s renormalization group.
Our findings provide (1) the first quantitative predictor for Riemann zero spacing based
on intrinsic properties, (2) a bridge between the Riemann Hypothesis and prime distribution
conjectures, (3) a thermodynamic framework unifying disorder and coherence effects, and (4)
concrete predictions for future computational and theoretical work. The universal state equation
establishes Lehmer pair spacing as a window into the deep arithmetic structure encoded in the
explicit formula.
close consecutive zeros of the Riemann zeta function—revealing a hierarchical structure connect-
ing phase disorder, prime pair correlations, and field-theoretic principles. Through systematic
analysis of the first 10,000 non-trivial zeros, we discover that spacing obeys a thermodynamic-like
relation at three progressive levels.
Level I (Base Equation): For Lehmer pairs in the platform region defined by prime phase
alignment, normalized spacing ∆γ follows ∆γ = ∆γ0 + κ · I, where I is a disorder parameter
constructed from phases ϕp(γ) = [γ log p/2π] mod 1 at the optimal prime set {5, 7, 11}. This
achieves R2 = 0.225 with p < 10−20 on n = 336 platform pairs.
Level II (Prime Pairs): Extending the model to include twin (k = 2), cousin (k = 4),
and sexy (k = 6) prime pair coherence terms dramatically improves predictive power to R2 =
0.335—a 56% increase. Twin primes alone contribute ∆R2 = +0.10 (46% improvement, p <
10−11). All three pair types exhibit negative coefficients, indicating that phase coherence reduces
spacing, establishing a quantitative link between the Hardy-Littlewood k-tuple conjecture and
zero distribution.
Level III (Field Theory): The hierarchical structure suggests an analogy to general
relativity, where the linear equation represents a “Newtonian limit” of a deeper field equation.
We observe scale dependence of the coupling constant κ (varying 21% across γ ranges) and
identify a mathematical cosmological constant Λ(math) ≈ π, hinting at fundamental connections
to quantum field theory’s renormalization group.
Our findings provide (1) the first quantitative predictor for Riemann zero spacing based
on intrinsic properties, (2) a bridge between the Riemann Hypothesis and prime distribution
conjectures, (3) a thermodynamic framework unifying disorder and coherence effects, and (4)
concrete predictions for future computational and theoretical work. The universal state equation
establishes Lehmer pair spacing as a window into the deep arithmetic structure encoded in the
explicit formula.
Martin Doina
Nov 18, 2025, 7:52:45 AMNov 18
to SeqFan
M F Hasler
Nov 18, 2025, 3:51:02 PMNov 18
to seq...@googlegroups.com
On Tue, Nov 18, 2025 at 8:52 AM Martin Doina <dhela...@gmail.com> wrote:
I take the approach in nonlinear , have a look it may be useful:
You write:
The true, fundamental zeta function is not the linear sum. It is the curved sum: (...something...)
But people are interested in the properties of Riemann's zeta function, not yours.
It does not really matter which one is the "true, fundamental" one.
People have linked Riemann's zeta function to many other properties.
These relations are rigorously established, but only for Riemann's zeta function,
not for your "curved" one, even if it were "better" or "the right one" in some sense.
Also, your "curvature aware addition"
a(+)b := arcsin( k · (a+b)) with k ~ 0.5599...
gives for example 0 (+) 1 = 0.594...
Is this intended? Doesn't this change / invalidate the meaning of 0?
This "addition" has no neutral element (unless you would change to k = 1),
so it shouldn't be called addition, which is a term reserved for abelian group operations,
according to universally agreed terminology.
- Maximilian
David desJardins
Nov 18, 2025, 3:54:54 PMNov 18
to seq...@googlegroups.com
This thread also doesn’t have much to do with sequences. Maybe it should be on a different list for recreational or research mathematics.
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