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Library · hunts/prime_zeta_rightmost/THEOREM.md

THEOREM.md: the replacement theorem (WP3, WP4)

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Hunt: prime_zeta_rightmost. Written 2026-08-16. This file states and proves the replacement for the two conjectures in OEIS A107311, against the verbatim text pinned in SOURCE.md section 2. Its numeric inputs are the decided enclosures in decided.json and theorem_inputs.json, each stated with backend and precision; everything else is proved here or cited with an access note (section 7). Vocabulary per MISSION.md: the strongest words used are measured, observed and decided (an enclosure with exact endpoint sign logic). Nothing in this file is a repo-level result until the case log in hunts/README.md says how the hunt ended.

The claims under attack, word for word as served (SOURCE.md section 2):

Conjecture 1: the real parts of the zeros of the prime zeta function are not greater than this constant. Conjecture 2: the real parts of the zeros of the anyone subset of the prime zeta function are not greater than this constant.

where "this constant" is x*, the real root of zeta(x) = 2. Both are false. Theorems A and B below replace Conjecture 1: the true supremum of the real parts is sigma_c = 1.77954465354699... > x* + 1/20, approached but not attained. Theorems C1 and C2 replace Conjecture 2: the subset {p >= 3} already has zeros beyond sigma_c, and tail subsets {p >= p_k} have thresholds growing without bound.

0. Setting, notation, and the decided inputs

Throughout, p ranges over primes. For a prime q let

S_q = {p prime : p >= q}, P_q(s) = sum_{p >= q} p^{-s},

so P_2 = P is the prime zeta function and P_3 is the {p >= 3} subset series. Each series converges absolutely and locally uniformly in Re s > 1 (comparison with sum n^{-sigma}), so P_q is analytic there. Every statement in this file concerns zeros with Re s > 1. By the reading-independence argument of SOURCE.md section 2, a zero with Re s > 1 is a zero of the defining series itself, where every reading of "the prime zeta function" (series, Glaisher continuation, any branch) agrees; so the refutations below hold under every reading at once.

The balance functions, on (1, infinity):

h_q(sigma) = sum_{p > q} p^{-sigma} - q^{-sigma}, g_q(sigma) = q^sigma * h_q(sigma) = sum_{p > q} (q/p)^sigma - 1.

Since q^sigma > 0, h_q and g_q have the same sign and the same roots. For q = 2 and q = 3 these are the functions bisected in decide.py:

h(sigma) = P(sigma) - 2^(1-sigma) = h_2(sigma), u(sigma) = P(sigma) - 2^(-sigma) - 2*3^(-sigma) = h_3(sigma).

sigma_c := sigma_c(2) and sigma_3 := sigma_c(3) denote the unique roots of h_2 and h_3 (uniqueness is Lemma 1 below; the decided derivative guards in decided.json confirm it independently on the bracketed ranges).

T1. Decided inputs

Backends: flint = python-flint (arb) at 350 bits; iv = mpmath.iv at dps 40. Both legs decided every row below.

From decided.json (2026-08-16):

x* in [1.7286472389981836181351030102976660, 1.7286472389981836181351030102977450] (flint, 350 bits) in [1.7286472389981835995, 1.7286472389981836884] (iv, dps 40) OEIS A107311's 102-digit value interval lies inside both backend intervals (decided); the flint interval lies inside the iv interval (decided). sigma_c in [1.7795446535469941164458987869654405, 1.7795446535469941164458987869655195] (flint, 350 bits) in [1.7795446535469636, 1.7795446535470547] (iv, dps 40) sigma_3 in [1.8252259560738457623878727108889264, 1.8252259560738457623878727108890054] (flint, 350 bits) in [1.8252259559929, 1.8252259560861] (iv, dps 40)

sigma_c - x* > 1/20 decided on both backends (exact rational compare of enclosure endpoints; the difference exceeds 0.0508974145 on the flint leg). h(x*) > 1/60 decided on both backends, with h evaluated over the whole x* enclosure (flint enclosure [0.016907377213856298105802854273997, 0.016907377213856298105802854274113]). endpoint signs h(17/10) > 0, h(19/10) < 0, u(9/5) > 0, u(37/20) < 0: decided on each backend before its bisection started (decided.json, bracket_endpoint_signs; a bracket with undecided endpoint signs raises, and no raise occurred). guards h'(s) < 0 on [1.7, 1.9] and u'(s) < 0 on [1.8, 1.85], decided on both backends; zeta'(x) < 0 on (1, 2) termwise-exactly.

From theorem_inputs.json (2026-08-16):

D1 h(7/4) in [0.009465637293734518587514821979513, 0.009465637293734518587514821979514] (flint, 350 bits; iv dps 40 confirms to 14 digits); h(7/4) > 1/128 decided on both. D2 u(9/5) in [0.004312491729413063275865581947004, 0.004312491729413063275865581947005] (flint, 350 bits; iv confirms); u(9/5) > 1/256 decided on both. D3 log2(69/(5 log 23)) in [2.137903503656002856060611381367, 2.137903503656002856060611381368] (flint, 350 bits; iv dps 40 agrees to all displayed digits);

17/8 decided on both.

W1 hi(x* enclosure) < 173/100 on both backends: any zero with Re s >= 1.73 has Re s > x*. W2 lo(sigma_c enclosure) > 7/4 on both backends. W3 lo(sigma_c enclosure) > 177/100 on both backends: the window (1.73, 1.77) sits inside (1, sigma_c). W4 hi(sigma_c enclosure) < 89/50 on both backends: any zero with Re s >= 1.78 has Re s > sigma_c. W5 lo(sigma_3 enclosure) > 91/50 on both backends: the window (1.78, 1.82) sits inside (1, sigma_3).

W1-W5 are exact Fraction comparisons against outward-rounded decimal endpoints; outward rounding only widens an enclosure, so each comparison also holds for the underlying enclosure. Everything else in this file is mathematics, proved below or cited in section 7.

1. Lemmas

Lemma 1 (the balance function and its unique root)

Let q be prime. On (1, infinity):

(i) the series defining g_q converges locally uniformly, and g_q is continuous; (ii) g_q is strictly decreasing; (iii) g_q(sigma) -> -1 as sigma -> infinity, and g_q(sigma) > 0 for some sigma > 1; (iv) hence g_q has exactly one root sigma_c(q) in (1, infinity), and

h_q > 0 on (1, sigma_c(q)), h_q(sigma_c(q)) = 0, h_q < 0 on (sigma_c(q), infinity).

Proof. (i) For [a, b] contained in (1, infinity) and sigma in [a, b], each term satisfies (q/p)^sigma <= (q/p)^a, and sum_{p > q} (q/p)^a <= q^a * sum_{n >= 2} n^{-a} < infinity; the Weierstrass M-test gives uniform convergence on [a, b], and a uniform limit of continuous functions is continuous.

(ii) For 1 < sigma < sigma', g_q(sigma) - g_q(sigma') = sum_{p > q} [(q/p)^sigma - (q/p)^{sigma'}]. Every term is strictly positive because 0 < q/p < 1, and the sum of a convergent series of positive terms is positive. So g_q(sigma) > g_q(sigma').

(iii) For sigma >= 2,

0 < g_q(sigma) + 1 = sum_{p > q} (q/p)^sigma <= sum_{n > q} (q/n)^sigma <= (q/(q+1))^sigma + q^sigma * int_{q+1}^infinity x^{-sigma} dx = (q/(q+1))^sigma * (1 + (q+1)/(sigma - 1)) -> 0

as sigma -> infinity, since q/(q+1) < 1. For positivity near 1: by Lemma 1a below, sum_{p > q} 1/p diverges, so there are finitely many primes q < p_1 < ... < p_J with q * sum_{j <= J} 1/p_j > 2. Then

liminf_{sigma -> 1+} g_q(sigma)

= lim_{sigma -> 1+} sum_{j <= J} (q/p_j)^sigma - 1

= q * sum_{j <= J} 1/p_j - 1 > 1 > 0,

so g_q(sigma) > 0 for sigma close enough to 1.

(iv) By (iii) there are points where g_q > 0 and points where g_q < 0 (any sigma large enough that g_q(sigma) < -1/2); by (i) and the intermediate value theorem g_q has a root; by (ii) the root is unique and the sign pattern is as stated. Multiplying by q^{-sigma} > 0 transfers the sign pattern to h_q. QED.

For q = 2 and q = 3 the decided endpoint signs in T1 (h(17/10) > 0, h(19/10) < 0; u(9/5) > 0, u(37/20) < 0) place the roots in (1.7, 1.9) and (1.8, 1.85), and the decided bisection brackets in T1 locate them to the stated widths. The bisection invariant maintained decided opposite endpoint signs at every step, so each reported bracket contains a root of the respective function; by Lemma 1(iv) that root is the only one, so the brackets locate sigma_c and sigma_3 themselves. (No monotonicity of h or u enters this location argument, only sign logic; the decided derivative guards are an independent confirmation on the bracketed ranges.)

Lemma 1a (Euler: the prime harmonic series diverges)

sum_p 1/p diverges; hence so does sum_{p > q} 1/p for every q.

Proof. For an integer N >= 2, every n <= N factors into prime powers with all primes <= N (fundamental theorem of arithmetic, cited in section 7), so expanding each geometric factor,

prod_{p <= N} (1 - 1/p)^{-1} = prod_{p <= N} (1 + 1/p + 1/p^2 + ...)

= sum_{n <= N} 1/n > log N.

Taking logarithms, sum_{p <= N} -log(1 - 1/p) > log log N. For 0 < x <= 1/2,

-log(1 - x) = x + x^2/2 + x^3/3 + ... <= x + (x^2/2) * (1 + x + x^2 + ...) = x + x^2/(2(1-x)) <= x + x^2,

so sum_{p <= N} 1/p > log log N - sum_p 1/p^2 > log log N - 1, which is unbounded in N. Removing the finitely many primes <= q changes the sum by a constant. QED.

Lemma 2 (polygon lemma)

Let r_1 >= r_2 >= ... >= r_N > 0 with N >= 2 and r_1 <= r_2 + ... + r_N. Then there are real phases theta_1, ..., theta_N with

sum_{j <= N} r_j * e^{i theta_j} = 0.

Proof. N = 2: the hypothesis gives r_1 <= r_2, and the ordering gives r_2 <= r_1, so r_1 = r_2; take (theta_1, theta_2) = (0, pi).

N >= 3: partition into three groups. Group a is {r_1}, with sum a = r_1. Distribute r_2, r_3, ..., r_N in this order, each to whichever of the two piles b, c currently has the smaller sum (ties arbitrarily). After placing r_2 the pile difference is |b - c| = r_2; inductively, placing r_j (which satisfies r_j <= r_2) onto the smaller pile changes the difference from d to |d - r_j| <= max(d, r_j) <= r_2. So at the end |b - c| <= r_2 <= r_1, and b + c = r_2 + ... + r_N >= r_1 by hypothesis, and b >= r_2 > 0, c >= r_3 > 0 (the first two placements land on empty piles). The three sums (a, b, c) therefore satisfy all three triangle inequalities:

a <= b + c (hypothesis), b <= a + c (since b - c <= r_2 <= a), c <= a + b (same).

Consequently |a - b| <= c <= a + b, which is equivalent to |a^2 + b^2 - c^2| <= 2ab, so

phi = arccos( (a^2 + b^2 - c^2) / (2ab) )

is well defined in [0, pi]. Then

|a + b e^{i(pi - phi)}|^2 = a^2 - 2ab cos(phi) + b^2 = c^2,

so a + b e^{i(pi - phi)} = c e^{i xi} for some real xi, and

a + b e^{i(pi - phi)} + c e^{i(xi + pi)} = 0.

Assign theta_j = 0 for the member of group a, theta_j = pi - phi for every member of group b, and theta_j = xi + pi for every member of group c; the grouped sum is the displayed one, so it vanishes. QED.

This is the classical closing-polygon fact ("a polygon with prescribed side lengths closes iff no side exceeds the sum of the others"); the proof is included so no citation carries load.

Lemma 3 (multiplicative independence of the primes)

Let q_1 < q_2 < ... < q_n be distinct primes and m_1, ..., m_n integers with q_1^{m_1} * ... * q_n^{m_n} = 1. Then every m_j = 0. Equivalently, {log q_1, ..., log q_n} is linearly independent over Q.

Proof. Move the factors with negative exponents to the other side:

prod_{m_j > 0} q_j^{m_j} = prod_{m_j < 0} q_j^{-m_j}.

Both sides are positive integers, and their prime supports are disjoint subsets of {q_1, ..., q_n}. By the uniqueness half of the fundamental theorem of arithmetic (section 7), an integer larger than 1 cannot have two factorizations with disjoint prime supports, so both sides equal 1 and all exponents vanish. For Q-linear independence: a rational relation sum c_j log q_j = 0 clears denominators to an integer relation sum m_j log q_j = 0, which exponentiates to prod q_j^{m_j} = 1. QED.

Lemma 4 (phase steering with positive density; Kronecker via Weyl)

Let q_1 < ... < q_n be distinct primes, alpha_1, ..., alpha_n real targets, and 0 < delta <= 1. Let

G = { t in R : |e^{-i t log q_j} - e^{i alpha_j}| <= delta for every j <= n }.

Then G has positive lower density: with |.| Lebesgue measure,

liminf_{T -> infinity} |G intersect [0, T]| / T >= (delta/(2 pi))^n.

In particular G is unbounded above, and for every spacing Delta > 0 it contains an increasing sequence t_1 < t_2 < ... with t_{k+1} - t_k >= Delta.

Proof. Since |e^{ia} - e^{ib}| = 2 |sin((a - b)/2)| <= dist(a - b, 2 pi Z), it suffices to steer angles: t belongs to G whenever

dist( -t log q_j - alpha_j , 2 pi Z ) <= delta for every j.

Work on the n-torus T^n = (R/Z)^n with coordinates

y_j(t) = ( -t log q_j - alpha_j ) / (2 pi) mod 1.

The angle condition says y(t) lies in the closed box B of points whose j-th coordinate is within delta/(2 pi) of 0 (mod 1), for every j; B has measure (delta/pi)^n. Let B' be the shrunk box with half-width delta/(4 pi) per coordinate, of measure (delta/(2 pi))^n, and let phi be the product of per-coordinate continuous tent functions equal to 1 on the B' interval, 0 outside the B interval, linear between; then 1_B >= phi >= 1_{B'} pointwise and int phi >= (delta/(2 pi))^n.

Fix eta > 0. By the Stone-Weierstrass theorem (trigonometric polynomials are dense in continuous functions on T^n, sup norm; section 7) there is a finite sum p_eta(y) = sum_{m in M} c_m e^{2 pi i m . y} with sup |phi - p_eta| <= eta. For the mean along the flow:

(1/T) int_0^T e^{2 pi i m . y(t)} dt, m . y(t) = -( t / (2 pi) ) * sum_j m_j log q_j

For m != 0 the coefficient omega_m = -(sum_j m_j log q_j)/(2 pi) of t is nonzero by Lemma 3, so the integral has modulus at most 2 / (2 pi |omega_m| T) -> 0 as T -> infinity. For m = 0 the integrand is the constant c_0, and |c_0 - int phi| = |int (p_eta - phi)| <= eta. Since M is finite,

(1/T) int_0^T phi(y(t)) dt

= (1/T) int_0^T Re p_eta(y(t)) dt - eta

-> Re c_0 - eta >= int phi - 2 eta.

Using 1_B >= phi,

liminf_T (1/T) |{ t in [0, T] : y(t) in B }|

= int phi - 2 eta >= (delta/(2 pi))^n - 2 eta,

and eta was arbitrary, so the lower density is at least (delta/(2 pi))^n. A set of positive lower density is unbounded (a subset of [0, T_0] has density 0), and greedily picking points separated by at least Delta from an unbounded set never terminates. QED.

Provenance: this is Kronecker's approximation theorem in its flow form, with Weyl's equidistribution method giving the density; sources in section 7. The proof is included so the citations carry no load.

Lemma 5 (equality in the triangle inequality)

Let (a_k) be complex numbers, countably many, with sum |a_k| finite and positive. If |sum a_k| = sum |a_k|, then there is a single unit complex number w with a_k = |a_k| * w for every k.

Proof. Let A = sum a_k and w = A / |A| (defined since |A| = sum |a_k| > 0). Then

sum |a_k| = |A| = Re( conj(w) * A ) = sum Re( conj(w) * a_k ),

so sum ( |a_k| - Re(conj(w) a_k) ) = 0 with every summand nonnegative (since Re z <= |z|). Hence Re(conj(w) a_k) = |a_k| for every k, which forces conj(w) a_k = |a_k|, that is a_k = |a_k| w. QED.

2. Theorem A: the wall

Theorem A. Let q be prime and sigma_c(q) the unique root from Lemma 1(iv). Then:

(a) for every s with Re s = sigma > sigma_c(q),

|P_q(s)| >= q^{-sigma} - sum_{p > q} p^{-sigma} = -h_q(sigma) > 0;

(b) P_q has no zero with Re s = sigma_c(q).

Hence every zero s of P_q with Re s > 1 satisfies Re s < sigma_c(q), and there are no zeros with Re s >= sigma_c(q).

In particular, for q = 2: every zero of the prime zeta function P with Re s > 1 has Re s < sigma_c, and there are no zeros with Re s >= sigma_c, where sigma_c is located by the decided enclosures of T1 (1.77954465354699411644589878696544... , flint 350 bits, iv dps 40 confirming). For q = 3 the same holds with sigma_3 (1.82522595607384576238787271088892... , same backends).

Proof. (a) By the triangle inequality, for Re s = sigma,

|P_q(s)| = | q^{-s} + sum_{p > q} p^{-s} |

= |q^{-s}| - sum_{p > q} |p^{-s}|

= q^{-sigma} - sum_{p > q} p^{-sigma} = -h_q(sigma),

and h_q(sigma) < 0 for sigma > sigma_c(q) by Lemma 1(iv).

(b) Suppose P_q(s) = 0 with s = sigma_c(q) + i t. Then q^{-s} = - sum_{p > q} p^{-s}, and taking moduli,

q^{-sigma_c(q)} = | sum_{p > q} p^{-s} | <= sum_{p > q} p^{-sigma_c(q)} = q^{-sigma_c(q)},

the last equality because h_q(sigma_c(q)) = 0. So the triangle inequality holds with equality for the infinite sum, and Lemma 5 gives a unit w with p^{-s} = p^{-sigma_c(q)} w for every p > q. Pick any three primes q < p1 < p2 < p3 (there are infinitely many primes). Equal phases mean

t * log(p2/p1) = 2 pi m, t * log(p3/p1) = 2 pi n

for some integers m, n. If t != 0 then m != 0 and n != 0 (the logarithms are nonzero); eliminating t,

n * log(p2/p1) = m * log(p3/p1), i.e. p2^n * p1^{m-n} * p3^{-m} = 1,

so by Lemma 3 m = n = 0, a contradiction. Hence t = 0; but then P_q(sigma_c(q)) = q^{-sigma_c(q)} + sum_{p > q} p^{-sigma_c(q)} = 2 q^{-sigma_c(q)} > 0, contradicting P_q(s) = 0. So no such zero exists. Combining (a) and (b): no zeros with Re s >= sigma_c(q); since sigma_c(q) > 1, every zero in Re s > 1 has Re s < sigma_c(q). QED.

Grades: Theorem A's proof is exact mathematics with no numeric input. The numerics enter only in locating sigma_c and sigma_3 (the T1 enclosures, decided on both backends) and in comparing them with x* (the decided separation), which is what turns Theorem A into a statement about the conjectures.

3. Theorem B: existence of zeros up to the wall

Theorem B. Let q be prime, and let sigma_1 and eps > 0 satisfy (sigma_1 - eps, sigma_1 + eps) contained in (1, sigma_c(q)). Then P_q has infinitely many zeros with Re s in (sigma_1 - eps, sigma_1 + eps); indeed infinitely many with |Re s - sigma_1| < eps/2, with imaginary parts unbounded above. (Coefficients are real, so zeros come in conjugate pairs.)

Proof. Write sigma_- = sigma_1 - eps/2. Note sigma_- > 1: the window hypothesis gives sigma_1 - eps >= 1, so sigma_- = sigma_1 - eps + eps/2

= 1 + eps/2 > 1.

Step 1 (ring margin). sigma_1 < sigma_c(q), so h_q(sigma_1) > 0 by Lemma 1(iv):

q^{-sigma_1} < sum_{p > q} p^{-sigma_1}.

No numerics are needed for general sigma_1; for the two demonstration instances used later the margin is also decided directly, on both backends: h_2(7/4) > 1/128 (T1, D1) and h_3(9/5) > 1/256 (T1, D2).

Step 2 (aggregated tail and the twisted series). Let T_X = sum_{p in S_q, p > X} p^{-sigma_1}, a tail of a convergent series, strictly positive (there are infinitely many primes) and decreasing to 0 in X. Choose X large enough that T_X <= q^{-sigma_1} and that some prime lies in (q, X]. Consider the finite list of positive reals

q^{-sigma_1}; p^{-sigma_1} for p in S_q, q < p <= X; T_X.

Its largest member is q^{-sigma_1} (every p-term with p > q is smaller; T_X <= q^{-sigma_1} by choice of X), and by Step 1

q^{-sigma_1} < sum_{q < p <= X} p^{-sigma_1} + T_X,

which is the sum of the other members. Lemma 2 applies (N >= 3) and gives phases theta_q, theta_p (q < p <= X), theta_* with

q^{-sigma_1} e^{i theta_q}

Set theta_p := theta_* for every p in S_q with p > X, and define the twisted series

P_theta(s) = sum_{p in S_q} e^{i theta_p} p^{-s}.

At s = sigma_1 the tail sums to e^{i theta_*} T_X, so P_theta(sigma_1) = 0 exactly.

Step 3 (the twisted series is analytic and not identically zero). The same M-test as Lemma 1(i) shows the series converges locally uniformly in Re s > 1, so P_theta is analytic there (locally uniform limits of analytic functions are analytic). It is not identically zero: the wall bound is phase-blind, so for Re s = sigma > sigma_c(q),

|P_theta(s)| >= q^{-sigma} - sum_{p > q} p^{-sigma} > 0

exactly as in Theorem A(a).

Step 4 (an isolating circle). The zeros of the analytic, not identically zero P_theta are isolated, so the compact disk |s - sigma_1| <= eps/2 (inside Re s > 1, since sigma_1 - eps/2 = sigma_- > 1) contains finitely many of them, giving finitely many moduli |z - sigma_1|. Choose

rho in (0, eps/2] with rho != |z - sigma_1| for every zero z,

so P_theta has no zero on the circle C = {|s - sigma_1| = rho}, and

m := min_{s on C} |P_theta(s)| > 0

(a continuous, nonvanishing modulus on a compact circle).

Step 5 (cutting the free tail). Every s on or inside C has Re s >= sigma_1 - rho >= sigma_-. Since sum_{p} p^{-sigma_-} converges, choose Y >= X with

2 * sum_{p in S_q, p > Y} p^{-sigma_-} <= m/2.

Step 6 (steering). Let C_Y = sum_{p in S_q, p <= Y} p^{-sigma_-} (finite, positive) and delta = min(1, m/(4 C_Y)). Apply Lemma 4 to the finitely many primes p in S_q with p <= Y, targets theta_p: the set G of times t_0 with

| p^{-i t_0} - e^{i theta_p} | <= delta for all p in S_q, p <= Y

has positive lower density.

Step 7 (comparison and Rouche). Fix t_0 in G. For s on C,

P_q(s + i t_0) - P_theta(s) = sum_{p in S_q} ( p^{-i t_0} - e^{i theta_p} ) p^{-s},

and splitting at Y, using |p^{-s}| = p^{-Re s} <= p^{-sigma_-} and |p^{-i t_0} - e^{i theta_p}| <= 2:

| P_q(s + i t_0) - P_theta(s) | <= delta * C_Y + 2 * sum_{p > Y, p in S_q} p^{-sigma_-} <= m/4 + m/2 < m <= |P_theta(s)|.

Both s -> P_q(s + i t_0) and P_theta are analytic on an open neighborhood of the closed disk bounded by C (the disk lies in Re s > 1, and translation by i t_0 preserves the half-plane). The strict inequality |f - g| < |g| on C (with f = P_q(. + i t_0), g = P_theta) is Rouche's hypothesis (section 7; it also forces g != 0 on C, consistent with Step 4). Therefore f and g have the same number of zeros inside C, counted with multiplicity; g has at least one (sigma_1, Step 2). So there is s' with |s' - sigma_1| < rho and P_q(s' + i t_0) = 0, i.e. a zero of P_q at s'' = s' + i t_0 with

Re s'' in (sigma_1 - rho, sigma_1 + rho) subset [sigma_1 - eps/2, sigma_1 + eps/2], Im s'' in (t_0 - rho, t_0 + rho).

Step 8 (infinitude). By Lemma 4, G contains t_1 < t_2 < t_3 < ... with consecutive gaps larger than 2 rho. The zeros produced for distinct t_k have imaginary parts in pairwise disjoint intervals, so they are pairwise distinct, and their imaginary parts tend to infinity. Every one of them has Re s within eps/2 of sigma_1, hence inside (sigma_1 - eps, sigma_1 + eps). QED.

Corollary B1 (the supremum). For every prime q,

sup { Re s : P_q(s) = 0, Re s > 1 } = sigma_c(q),

and the supremum is not attained. Proof. Theorem A gives Re s < sigma_c(q) for every zero. For any sigma_1 < sigma_c(q) choose eps with (sigma_1 - eps, sigma_1 + eps) inside (1, sigma_c(q)); Theorem B gives zeros with Re s > sigma_1 - eps, and sigma_1 was arbitrary below sigma_c(q). QED.

Corollary B2 (Conjecture 1 is false). Take q = 2, sigma_1 = 7/4, eps = 1/50. The window (1.73, 1.77) lies inside (1, sigma_c): its right end is below sigma_c because lo(sigma_c enclosure) > 177/100 (decided, W3, both backends), and its left end exceeds 1. Theorem B: P has infinitely many zeros with Re s in (1.73, 1.77). Every such zero has Re s > x*, because hi(x* enclosure) < 173/100 (decided, W1, both backends). Hence the real parts of the zeros of the prime zeta function are not bounded by x*: Conjecture 1 of OEIS A107311 is false. By Corollary B1 the correct bound is sigma_c, which exceeds x* by more than 1/20 (decided separation, T1), and it is a supremum, not a maximum.

4. Theorem C: subsets

Theorem C1 (the subset {p >= 3} refutes Conjecture 2)

Theorems A and B hold verbatim for q = 3 (their proofs never use q = 2: Lemmas 1-5 are stated for a general prime q, and the independence in Lemmas 3 and 4 holds for any finite set of distinct primes). Hence:

(a) every zero of P_3(s) = sum_{p >= 3} p^{-s} with Re s > 1 has Re s < sigma_3, and none has Re s >= sigma_3; (b) taking sigma_1 = 9/5, eps = 1/50: the window (1.78, 1.82) lies inside (1, sigma_3) because lo(sigma_3 enclosure) > 91/50 (decided, W5, both backends), and the ring margin at 9/5 is decided directly (u(9/5) > 1/256, D2). P_3 has infinitely many zeros with Re s in (1.78, 1.82).

Every zero from (b) has Re s > 1.78 > 1.73 > x* (decided, W1). So the subset S = {p >= 3} violates Conjecture 2 of OEIS A107311: its zeros' real parts are not bounded by x*.

Remark (a subset out-walls the full series). By W4 (hi(sigma_c enclosure) < 89/50, decided, both backends), every zero from (b) also has Re s > sigma_c: removing the prime 2 moves the wall right, and the subset series has zeros strictly beyond the supremum for the full series. The decided enclosures give sigma_3 - sigma_c > 0.045 (exact rational compare of lo(sigma_3) - hi(sigma_c) on the flint leg gives at least 1.8252259560738457... - 1.7795446535469942 > 0.0456813); the subset conjecture is not merely false, it fails by more than the full-series one.

Theorem C2 (tail subsets have unbounded walls)

For the k-th prime p_k let S_{p_k} = {p : p >= p_k} and let sigma_c(p_k) be its wall (Lemma 1(iv), which applies to every q since Lemma 1a gives the divergence for every tail). Then for every p_k >= 23,

sigma_c(p_k) >= log2( 3 p_k / (5 log p_k) ),

and the right side tends to infinity with k. Consequently, for every real M there is a k such that P_{p_k} has infinitely many zeros with Re s > M.

Proof. The count input is Rosser and Schoenfeld's Corollary 3 of their Theorem 2, inequality (3.8) (section 7):

3x / (5 log x) < pi(2x) - pi(x) for 20.5 <= x.

Apply it at x = p_k >= 23 > 20.5: the interval (p_k, 2 p_k] contains more than 3 p_k / (5 log p_k) primes, each a member of S_{p_k} exceeding p_k and at most 2 p_k. Hence for any T > 1,

sum_{p > p_k} p^{-T} >= sum_{p in (p_k, 2 p_k]} p^{-T}

= (3 p_k / (5 log p_k)) * (2 p_k)^{-T}.

Since (2 p_k)^{-T} = 2^{-T} p_k^{-T}, the right side exceeds p_k^{-T} exactly when 2^{-T} * 3 p_k / (5 log p_k) > 1, that is when

T < B_k := log2( 3 p_k / (5 log p_k) ).

So h_{p_k}(T) > 0 for every T in (1, B_k). If sigma_c(p_k) were smaller than B_k, any T in (sigma_c(p_k), B_k) would have h_{p_k}(T) < 0 by Lemma 1(iv), a contradiction; hence sigma_c(p_k) >= B_k. (When B_k <= 1 the inequality is trivially true since sigma_c(p_k) > 1.)

Unboundedness: x / log x is increasing for x > e (its derivative is (log x - 1)/log^2 x) and unbounded, and p_k -> infinity, so B_k -> infinity. Decided instance, k = 9, p_9 = 23:

B_9 = log2(69 / (5 log 23)) in [2.137903503656002856060611381367, 2.137903503656002856060611381368] (flint, 350 bits; mpmath.iv at dps 40 agrees to all displayed digits), B_9 > 17/8 decided on both backends (D3):

the wall of {p >= 23} exceeds 2.125, already far beyond x* and sigma_c.

Zero production: given M, choose k with B_k > max(M, 3/2) (possible since B_k -> infinity), so sigma_c(p_k) > max(M, 3/2). Set

sigma_1 = ( max(M, 3/2) + sigma_c(p_k) ) / 2, eps = ( sigma_c(p_k) - max(M, 3/2) ) / 4;

then (sigma_1 - eps, sigma_1 + eps) is contained in (max(M, 3/2), sigma_c(p_k)), which lies inside (1, sigma_c(p_k)). Theorem B for q = p_k gives infinitely many zeros of P_{p_k} with Re s > max(M, 3/2) >= M. QED.

Corollary C3 (Conjecture 2 fails without bound)

For every real M there is a subset S of the primes (a tail {p >= p_k}) such that sum_{p in S} p^{-s} has infinitely many zeros with Re s > M. No constant, in particular not x*, bounds the real parts of the zeros over all subsets. Conjecture 2 is false for every possible replacement constant, not just for x*.

Scope note: the refuting subsets are infinite. Nothing in this file is claimed about finite subsets (whose series are exponential polynomials with walls of their own); the conjecture quantifies over all subsets, so the infinite witnesses settle it.

5. What exactly is refuted

Conjecture 1 ("the real parts of the zeros of the prime zeta function are not greater than [x*]"): false. Theorem B (Corollary B2) produces infinitely many zeros of P with Re s in (1.73, 1.77), every one exceeding x* (decided compares W1, W3). Theorem A plus Corollary B1 identify the true threshold: sup of the real parts is sigma_c = 1.779544653546994... (decided enclosures, T1), strictly above x* by more than 1/20 (decided separation), approached but not attained. The zeros produced lie in Re s > 1, where every reading of "the prime zeta function" agrees with the series (SOURCE.md section 2), so the refutation is reading-independent.

Conjecture 2 ("... the anyone subset ..."): false, twice over. First, the concrete subset {p >= 3}: infinitely many zeros with Re s in (1.78, 1.82) (Theorem C1), beyond x* and even beyond sigma_c. Second, the tails {p >= p_k}: walls at least log2(3 p_k/(5 log p_k)) -> infinity (Theorem C2), with zeros beyond every fixed bound (Corollary C3). The {p >= 23} instance is decided: its wall exceeds 17/8 (D3).

What is not claimed. No explicit zero is exhibited; existence is by Rouche, and the first witnesses may sit at astronomical heights (MISSION.md; the value margins are parts in a thousand). Nothing is claimed about zeros with Re s <= 1, about the continuation beyond the series' half-plane, or about finite subsets. None of this bears on the Riemann Hypothesis: the zeros produced are zeros of P and of subset series in Re s > 1, not zeros of zeta, and x* keeps its correct role as the partial-sums threshold of Borwein, Fee, Ferguson and van der Waall (SOURCE.md section 3). The error in the OEIS comment is a transplant: a constant from a family whose wall is never attained (partial sums, with rationally dependent frequencies log n) was conjectured onto a family with independent frequencies, where the wall is different, larger, and actually approached.

6. Which step carries which grade

Per the vocabulary contract in MISSION.md, the replacement theorem is a composite: decided inequalities glued by classical arguments, with the glue proved in this file. The grade of each step:

Decided (interval/ball enclosure with exact endpoint sign logic; python-flint (arb) at 350 bits and mpmath.iv at dps 40, independent code paths, every item decided on both backends):

Proved in this file (complete proofs, no numeric content): Lemmas 1, 1a, 2, 3, 4, 5; Theorems A, B, C1, C2; Corollaries B1, B2, C3.

Cited without reproof (textbook or classical, hypotheses checked explicitly where used; access notes in section 7): the fundamental theorem of arithmetic (Lemmas 1a, 3); Rouche's theorem (Theorem B, step 7); Stone-Weierstrass on the torus (Lemma 4); Rosser-Schoenfeld Corollary 3, inequality (3.8) (Theorem C2); and standard facts of analysis (M-test, intermediate value theorem, analyticity of locally uniform limits, isolation of zeros of analytic functions).

Composite grade: every numeric input is decided on both backends; no claim rests on a measured-only number. The remaining steps are classical mathematics, proved here or standard. Nothing in this file is kernel-checked, and nothing in this file uses or may use the reserved enclosure vocabulary of zeta/rigor.py.

7. Citations, with access notes

  1. J. B. Rosser and L. Schoenfeld, "Approximate formulas for some functions of prime numbers", Illinois J. Math. 6 (1962), no. 1, pp. 64-94. Used: Corollary 3 of Theorem 2, inequality (3.8): 3x/(5 log x) < pi(2x) - pi(x) for 20.5 <= x. Access note (2026-08-16): a scanned copy served at denise.vella.chemla.free.fr /Rosser-Schoenfeld-1962.pdf was fetched and its text extracted; the corollary block reads "COROLLARY 3. We have 3x/(5 log x) < pi (2x) - pi (x) (3.8) ... for 20 1/2 <= x" (OCR of the scan; Greek pi transcribed). The neighboring Corollary 1 in the same extraction carries x/log x < pi(x) for 17 <= x and pi(x) < 1.25506 x/log x for 1 < x, the constants universally quoted from this paper, which cross-checks the OCR.
  2. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008. Used: the fundamental theorem of arithmetic (Theorem 2 there) for Lemmas 1a and 3; Chapter XXIII (Kronecker's theorem) and Section 22.7 (Mertens) as provenance for Lemmas 4 and 1a. Access note: theorem and chapter numbers quoted from memory, not re-checked against a copy in this session; no load rests on the numbering, since the proofs used here are included above.
  3. H. Weyl, "Uber die Gleichverteilung von Zahlen mod. Eins", Math. Ann. 77 (1916), pp. 313-352. Provenance for the equidistribution method in Lemma 4; the flow argument used is the standard Weyl-criterion computation, written out in full above. Access note: not consulted in this session.
  4. Rouche's theorem, in the form: f and g analytic on an open set containing a closed disk, |f - g| < |g| on the boundary circle; then f and g have equally many zeros (with multiplicity) inside. Standard texts: E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford, 1939, Chapter III (argument principle and Rouche); L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, 1979, Chapter 4, Section 5. Access note: section placements from memory, not re-checked in this session; the hypotheses are checked explicitly in Theorem B, step 7.
  5. Stone-Weierstrass theorem (trigonometric polynomials dense in the continuous functions on the n-torus, sup norm), e.g. W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, 1976, Chapter 7 (the complex, self-adjoint form). Access note: chapter from memory, not re-checked in this session.
  6. H. Bohr, "Zur Theorie der allgemeinen Dirichletschen Reihen", Math. Ann. 79 (1918), pp. 136-156. Provenance only: the vertical-translation method behind Theorem B descends from Bohr's work on general Dirichlet series; no statement of his is invoked. Access note: not consulted in this session.
  7. P. Borwein, G. Fee, R. Ferguson, A. van der Waall, "Zeros of Partial Sums of the Riemann Zeta Function", Experimental Mathematics 16 (2007), no. 1. Context only (the provenance of x*); pinned with access notes and two independent secondary quotes in SOURCE.md sections 1 and 3. No step of Theorems A-C depends on it.
  8. S. M. Gonek and A. H. Ledoan, "Zeros of partial sums of the Riemann zeta-function", IMRN 2010, no. 10, pp. 1775-1791; D. J. Platt and T. S. Trudgian, "Zeroes of partial sums of the zeta-function", LMS J. Comput. Math. 19 (2016). The secondary sources for the partial-sums wall, quoted verbatim in SOURCE.md section 3; context only.
  9. C.-E. Froberg, "On the prime zeta function", BIT 8 (1968), no. 3, pp. 187-202. The only prior literature found on zeros of P (four observed roots, Table I); pinned via the zbMATH review in SOURCE.md section 4. Nothing above uses it.
  10. OEIS A107311, revision 55 (2024-12-29), fetched 2026-08-15 and 2026-08-16 (byte-identical): the conjecture text under refutation and the 102 digits of x*, pinned byte-for-byte in SOURCE.md section 1.

8. Gaps and honesty notes