Lift Paper — F(u) · Modular Rings

Modular Lifting Rings
and Prime Geometry

The Analytic Lift Function · Constants · Zero Spectrum · Möbius

Wessen Getachew

math.NT · 2025–2026

Contents

0.The Setting
1.Definition of F(u)
2.Landmarks on [0, 4)
3.The Zero Spectrum
4.Derivative at u=1 · Constant S
5.Constants — Formal Names and Attribution
6.Algebraic Relations Among the Constants
7.Gap Ratio Theorem
8.φ Spectral Obstruction
8b.F(u) and Möbius
9.Open Problems ▾
10.Twin Primes in the Framework
Ref.Bibliography

Section 0

The Setting

Fix a positive integer M. The coprime residues of M are the elements of (ℤ/Mℤ)* — all r in {1,…,M} with gcd(r, M) = 1. A residue r lifts from ring M to ring M+1 if it satisfies gcd(r, M) = 1 and gcd(r, M+1) = 1 simultaneously. The long-run fraction of residues that lift converges to a constant C, defined by an Euler product over all primes.

The analytic lift function F(u) is the natural one-parameter generalization of this product. It unifies several of the most important constants in the framework — ζ(2), C, and the number 1 — as values at specific arguments, and it encodes the entire prime sequence in its zero set.

Section 1

Definition of F(u)

Definition 1.1 — The Analytic Lift Function

For u ∈ ℝ, define

F(u) = ∏_{p prime} (p² − u) / (p² − 1)

The product converges absolutely for all u ∈ ℝ \ {p² : p prime}. At u = p² for any prime p, F vanishes because the factor for that prime is zero. F is a meromorphic function of a complex variable u, real-valued and smooth on ℝ away from its zero set.

The factor at prime p is (p² − u)/(p² − 1). Its numerator is linear in u; its denominator is the fixed constant p² − 1 > 0. Each factor equals 1 at u = 1, equals p²/(p²−1) at u = 0, and equals (p²−2)/(p²−1) at u = 2.

Remark 1.2 — Convergence

For fixed u not a prime square, the product converges absolutely because log|(p²−u)/(p²−1)| ~ −u/p² for large p, and ∑ 1/p² converges. The convergence is uniform on compact subsets of ℝ avoiding the zero set.

Section 2

Landmarks on [0, 4) Algebraic

The interval [0, 4) is the maximal interval on which F is positive. It contains four distinguished points, each corresponding to a fundamental object.

Theorem 2.1 — Three Exact Values

(i) F(0) = ζ(2) = π²/6.
At u = 0, each factor is p²/(p²−1). The product ∏_p p²/(p²−1) is the Euler product for ζ(2). This is exact and classical.

(ii) F(1) = 1 exactly.
At u = 1, each factor is (p²−1)/(p²−1) = 1. The product of ones is 1. This holds term by term and requires no computation.

(iii) F(2) = C ≈ 0.530711806246.
At u = 2, each factor is (p²−2)/(p²−1). The product is the lift survival constant C, the long-run fraction of coprime residues that lift from ring M to M+1.

Proof

(i) By definition ζ(2) = ∑ n⁻² = ∏_p (1 − p⁻²)⁻¹ = ∏_p p²/(p²−1). This is the Euler product identity. (ii) Each factor at u=1 is (p²−1)/(p²−1) = 1, so F(1) = ∏_p 1 = 1. (iii) Substituting u=2 into the definition of F gives the Euler product for C by definition. □

Theorem 2.2 — F is Strictly Decreasing on [0, 4)

F is strictly decreasing on [0, 4). In particular, F(0) > F(1) > F(2) > F(3) > F(4) = 0, giving the chain ζ(2) > 1 > C > F(3) > 0.

Proof

The log-derivative d/du log F(u) = ∑_p −1/(p²−u) is strictly negative for u < 4 (all denominators p²−u are positive). Hence F is strictly decreasing on (−∞, 4). □

Figure 1. F(u) on [0, 4]. The four landmarks ζ(2), 1, C, and 0 are marked. F is strictly decreasing, positive on (−∞,4), and vanishes at u = 4 = 2².

Observation 2.3 — Arithmetic of the Landmark Values

The three nonzero landmarks satisfy:

F(0) · F(2) = ζ(2) · C = ζ(2) · ∏_p(1 − 2/p²)
C = ζ(2)·∏(1−2/p²) is established in §6 (Theorem 6.2) by term-by-term cancellation.

Also: F(0)/F(1) = ζ(2) and F(1)/F(2) = 1/C, so the three landmarks are in ratio ζ(2) : 1 : C with F(1) = 1 as the normalization point.

Section 3

The Zero Spectrum

Theorem 3.1 — The Zero Set of F

The zeros of F on ℝ are exactly the set {p² : p prime} = {4, 9, 25, 49, 121, 169, 289, …}. Each zero is simple (of order 1). The map p ↦ p² is a bijection from the primes to the zeros of F. Equivalently, √(zero) recovers the prime.

Proof

F(u) = 0 iff at least one factor is zero. The factor for prime p vanishes iff p² − u = 0, i.e., u = p². All prime squares are distinct (since primes are distinct), so each u = p² kills exactly one factor while leaving all others nonzero. Hence F(p²) = 0 and the zero has order exactly 1 (the factor is linear in u near u = p²). □

Theorem 3.2 — Sign Alternation

Let p₁ < p₂ < … be the primes in order. On the interval (p_n², p_{n+1}²), the sign of F is (−1)ⁿ. That is:

sign F(u) = (−1)^#{primes q : q² < u}
F > 0 on (−∞, 4), F < 0 on (4, 9), F > 0 on (9, 25), and so on.

The sign changes of F on ℝ occur at exactly the prime squares, and reading off the sign-change locations recovers the primes.

Proof

On (p_n², p_{n+1}²), exactly n factors have negative numerator (those with q ≤ p_n, so q² < u). All other factors are positive. The product of n negative factors and infinitely many positive ones has sign (−1)ⁿ. □

Interval	Zero bounding it	Sign of F	n
(−∞, 4)	2² = 4	+ (positive)	0
(4, 9)	3² = 9	− (negative)	1
(9, 25)	5² = 25	+ (positive)	2
(25, 49)	7² = 49	− (negative)	3
(49, 121)	11² = 121	+ (positive)	4
(121, 169)	13² = 169	− (negative)	5

Observation 3.3 — The Boundary of the Positive Region

The interval (−∞, 4) is the maximal interval on which F is positive. Its right endpoint is the square of the smallest prime, 2² = 4. All of the framework's constants — ζ(2), C, J, D₀, T — correspond to F evaluated at u ∈ [0, 2] ⊂ (−∞, 4). The prime 2 is not just the first prime: it is the boundary condition for the entire positive regime of F.

Section 4

The Derivative at u = 1

Since F(1) = 1, the derivative F'(1) is the slope of F at its unique normalization point. It introduces a new constant into the framework.

Theorem 4.1 — The Slope Constant S

F'(1) = −S where S = ∑_{p prime} 1/(p² − 1)
S ≈ 0.551693183169550. F'(1) is negative, consistent with F decreasing.

Proof

Differentiating log F(u) = ∑_p log((p²−u)/(p²−1)) with respect to u:

d/du log F(u) = ∑_p −1/(p²−u)

So F'(u)/F(u) = −∑_p 1/(p²−u). At u = 1: F'(1)/F(1) = −∑_p 1/(p²−1). Since F(1) = 1, we get F'(1) = −S where S = ∑_p 1/(p²−1). □

Definition 4.2 — The Constant S

S = ∑_{p prime} 1/(p²−1) = ∑_p 1/((p−1)(p+1)) ≈ 0.551693183169550.

S also satisfies the decomposition S = D₀ + Q, where:

Q = ∑_p 1/(p²(p²−1)) ≈ 0.099445877615934
Proof: 1/(p²−1) = 1/p² + 1/(p²(p²−1)). Algebraic, term by term.

Section 5

Constants of the Framework — Formal Names and Attribution

Eight constants appear in the framework. Each is a convergent product or sum over primes. The table below gives the formal name, definition, value, provenance, and OEIS reference for each. Constants marked ★ are introduced here in the context of the lift function and do not appear to have been previously named or studied as primary objects; they carry their formal definitions and require no ad hoc nomenclature. All values verified to 50,000 primes.

Notation

P(s) = ∑_{p prime} p^−s denotes the prime zeta function, introduced by Glaisher (1891) and studied systematically by Fröberg (1968). The Riemann zeta function is ζ(s) = ∑_n≥1 n^−s. The Feller–Tornier constant is C_FT = ½(1 + ∏_p(1 − 2/p²)), introduced in Feller & Tornier (1933).

ζ(2)

π²/6 = ∏_p p²/(p²−1)

1.644934066848226…

F(0). Riemann zeta function at 2. Euler (1734). OEIS A013661.

∏_p (p²−2)/(p²−1)

computing…

F(2). Lift survival constant. C = ζ(2)·(2C_FT−1). OEIS A065469.

C_FT

½(1 + ∏_p(1 − 2/p²))

computing…

Feller–Tornier constant. OEIS A065493. Feller & Tornier (1933).

P(2)

∑_p 1/p²

computing…

Prime zeta function at 2. OEIS A085548. Glaisher (1891).

∑_p 1/(p(p−1))

computing…

Sum of prime reciprocal differences. J = P(2) + T. OEIS A136141.

∑_p 1/(p²(p−1)) = J − P(2)

computing…

First-order carry of J over P(2).

S ★

∑_p 1/(p²−1) = ∑_k≥1 P(2k) = −F′(1)

computing…

Prime zeta series sum; derivative of F at 1. S = P(2) + Q.

Q ★

∑_p 1/(p²(p²−1)) = S − P(2)

computing…

Second-order carry of S over P(2).

U ★

∑_p 1/(p(p−1)(p+1)) = J − S

computing…

Gap between J and S.

Remark 5.1 — On nomenclature

The constants ζ(2), C_FT, and P(s) carry standard names and are not renamed here. The constant C = ∏(p²−2)/(p²−1) is recorded in OEIS A065469 and follows from Feller–Tornier theory; the lift interpretation is new. The constants S, Q, U are defined by convergent prime series; S equals the sum ∑_k≥1 P(2k) of prime zeta values, a classical identity, but its role as −F′(1) and the constants Q and U as carry terms are not found in prior literature. The notation J for ∑ 1/(p(p−1)) follows usage in Finch (2003). The ★ marker indicates constants whose properties in the present context are first developed here.

Section 6

Algebraic Relations Among the Constants

The following four identities hold term by term over all primes, with no appeal to limits. Each is a consequence of partial fraction decomposition applied to a single prime, then summed. They collectively show that the eight constants of §5 are not independent but are four views of a single arithmetic structure anchored at P(2).

Theorem 6.1 — Decomposition of J

For every prime p:

1/(p(p−1)) = 1/p² + 1/(p²(p−1))

Summing over all primes: J = P(2) + T. The left side is ∑ 1/(p(p−1)); the right side splits it into the prime zeta value P(2) = ∑ 1/p² and the carry term T = ∑ 1/(p²(p−1)).

Theorem 6.2 — The Lift Survival Identity

C = ζ(2) · ∏_p(1 − 2/p²) = ζ(2) · (2C_FT − 1)
Proof: ∏[p²/(p²−1)] · ∏[(p²−2)/p²] = ∏[(p²−2)/(p²−1)] = C. The factor p² cancels termwise. The Feller–Tornier relation C_FT = ½(1+∏(1−2/p²)) then gives the second equality.

This connects F(0) = ζ(2), F(2) = C, and the Feller–Tornier constant C_FT in a single identity. The three quantities were defined independently; the identity is not trivial.

Theorem 6.3 — Decomposition of S

For every prime p:

1/(p²−1) = 1/p² + 1/(p²(p²−1))

Summing: S = P(2) + Q. An equivalent form follows from expanding each factor as a geometric series: 1/(p²−1) = ∑_k≥1 p^−2k, so S = ∑_k≥1 P(2k). This expresses S as the sum of all even prime zeta values.

Theorem 6.4 — The Gap Identity

For every prime p:

1/(p(p−1)) − 1/(p²−1) = 1/(p(p−1)(p+1))

Summing: J − S = U, where U = ∑_p 1/(p(p−1)(p+1)).

The four identities together give a closed lattice:

J = P(2) + T
S = P(2) + Q
J = S + U
C = ζ(2) · (2C_FT − 1)
P(2) is the base; T, Q, U are the three carry terms at different algebraic levels.

Remark 6.5 — Independence from limit arguments

These identities hold for each prime individually before any summation takes place. The summed form follows by linearity. Numerical residuals at 50,000 primes lie below 10⁻¹³, consistent with floating-point precision only.

Section 7

Gap Ratio Theorem

For M = p# (primorial), define A(g, M) = #{r ≤ M : gcd(r,M)=1, gcd(r+g,M)=1}. For even gaps g₁, g₂ with odd prime factors inside M:

A(g₁,M)/A(g₂,M) = ∏_{q|g₁,q∤g₂}(q−1)/(q−2) · ∏_{q|g₂,q∤g₁}(q−2)/(q−1)
Independent of M. In particular A(6,M)/A(2,M) = 2 exactly at every primorial ≥ 6.

M	A(2,M)	A(6,M)	Ratio
6	1	2	2.0000 ✓
30	3	6	2.0000 ✓
210	15	30	2.0000 ✓
2310	135	270	2.0000 ✓

Section 8

φ as Spectral Obstruction

The jump spectrum is {1/(p(p−1)) : p prime}. Recovery formula: p(s) = (1+√(1+4/s))/2.

At s=1: 1+4/s = 5 → p = (1+√5)/2 = φ
√5 irrational ⟹ no prime at s=1. φ is the unique irrational obstruction in the spectrum.

For every real prime p, 1+4/s_p = (2p−1)² — always a perfect square. The golden ratio occupies the one position where this fails.

Section 8b

F(u) and Möbius

F(u) is not a Möbius inversion in the technical sense, but the Möbius function μ is embedded in F as its u=1 specialization. The key identity:

∏_p(1 − u/p²) = Σ_{n squarefree} (−u)^ω(n)/n²
where ω(n) = number of distinct prime factors. At u=1: weights (−1)^{ω(n)} = μ(n), so the sum becomes Σμ(n)/n² = 1/ζ(2).

Therefore F(1) = ζ(2)·(1/ζ(2)) = 1. The normalization is not coincidence — it is a Möbius cancellation. F(u) is a one-parameter family of Dirichlet sums; the Möbius case is u=1.

Section 9

Open Problems

Five open problems — click to expand

Does F appear as an operator? If det(I−uL)=F(u) for some L on a Hilbert space, primes are eigenvalues. What is L?

F(u) for complex u. Growth order, Hadamard product, relation to ζ(s) in the critical strip?

U ≈ 0.2215 geometrically. U = J−S = Σ1/(p(p−1)(p+1)). Clear algebraic origin, no geometric meaning in the lift framework yet.

Σ|E(n,N)| = O(N^1/2+ε)? Farey sector error sum. If yes, computational evidence for Franel–Landau RH equivalence.

F⁻¹(1/2) ∈ (2,4). Does it have arithmetic significance? Connection to Re(s)=1/2?

Section 10

Twin Primes in the Lift Framework Observation

The framework has two precise points of contact with twin primes. Neither proves the Twin Prime Conjecture. Both say something exact.

Theorem 10.1 — Exact gap ratios at primorials

For M = p# with p ≥ 5, the coprime pair counts satisfy:

A(2, M) = A(4, M) [twin pairs = cousin pairs, exactly]
A(6, M) / A(2, M) = 2 [sexy pairs = twice twin pairs, exactly]

The Hardy–Littlewood ratios S(4)/S(2) = 1 and S(6)/S(2) = 2 are usually stated asymptotically. These are exact integer identities at every primorial.

Proof

By CRT, A(g, M) = M · ∏_{q|M, q|g}(q−1)/q · ∏_{q|M, q∤g}(q−2)/q. For g=2 and g=4: both are even, no odd prime divides one but not the other, so every factor is the same — the products are equal. For g=6 vs g=2: only q=3 differs (3|6, 3∤2), contributing (3−1)/3 = 2/3 vs (3−2)/3 = 1/3. Ratio = (2/3)/(1/3) = 2. □

M	A(2,M) twin	A(4,M) cousin	A(6,M) sexy	Equal?	Ratio 6/2
30	3	3	6	✓	2
210	15	15	30	✓	2
2310	135	135	270	✓	2
30030	1485	1485	2970	✓	2

Remark 10.2 — Scope

The gap ratio identity shows that twin and cousin pairs have equal density at every primorial ring — an exact finite result. It does not imply infinitely many twin primes. That requires survival through all primorials simultaneously, which is precisely the difficulty of the conjecture. The finite result stands on its own.

References

Bibliography

[1] Euler, L. (1734). De summis serierum reciprocarum. Comment. Acad. Sci. Petropol. 7, 123–134. [Basel problem, ζ(2) = π²/6.]

[2] Feller, W. & Tornier, E. (1933). Mengentheoretische Untersuchung von Eigenschaften der Zahlenreihe. Math. Ann. 107, 188–232. [Feller–Tornier constant C_FT, OEIS A065493.]

[3] Glaisher, J. W. L. (1891). On the sums of inverse powers of the prime numbers. Quart. J. Math. 25, 347–362. [Prime zeta function P(s), OEIS A085548.]

[4] Fröberg, C.-E. (1968). On the prime zeta function. BIT 8, 187–202. [Systematic study of P(s); convergence and computation.]

[5] Finch, S. R. (2003). Mathematical Constants. Cambridge University Press. §2.2 (Meissel–Mertens), §2.4 (Feller–Tornier). [J = ∑ 1/(p(p−1)) tabulated; C_FT given with references.]

[6] Hardy, G. H. & Wright, E. M. (1979). An Introduction to the Theory of Numbers, 5th ed. Oxford University Press. [Euler product for ζ(2); Möbius inversion; prime distribution.]

[7] Franel, J. (1924). Les suites de Farey et le problème des nombres premiers. Göttinger Nachrichten 198–201. Landau, E. (1924). Bemerkungen zu der vorstehenden Abhandlung. Göttinger Nachrichten 202–206. [Franel–Landau RH equivalence via Farey sequences.]

[8] Hardy, G. H. & Littlewood, J. E. (1923). Some problems of Partitio Numerorum III. Acta Math. 44, 1–70. [Singular series S(k) for prime k-tuples; gap ratios in the primorial limit.]

[9] Weisstein, E. W. MathWorld, Wolfram Research. Entries: Feller-Tornier Constant; Prime Zeta Function; Basel Problem. [Numerical values and cross-references.]

[10] OEIS Foundation. The On-Line Encyclopedia of Integer Sequences. A013661 (ζ(2)); A065469 (C); A065493 (C_FT); A085548 (P(2)); A136141 (J).