# A weight-two spectral bound for the integrated theta tail Historical companion snapshot — September 15, 2026. The mathematical text and review-status statements below describe that snapshot, not the current consolidation status. Current scope and limitations: https://arithmeticconnectome.com/robin Internal source and audit filenames below identify private provenance records; they are not public downloads. The reviewed circulation PDF is available at: https://arithmeticconnectome.com/papers/robin-circulation.pdf Contact: research@arithmeticconnectome.com September 15, 2026. Standalone written lemma extracted from the reviewed analytic chain. This note is intended for focused mathematical and attribution review. It does not assert novelty, a numerical cutoff or Lean verification. ## 1. Input contract and conclusion All logarithms are natural. Let theta(x)=sum_(p<=x)log p and psi(x)=sum_(p^k<=x)log p, with k>=1. Let rho=beta+i gamma run over the nontrivial zeros of the ordinary Riemann zeta function with multiplicity. Fix the following inputs once, independently of y and all moving parameters. **(A1) Zeros and zero-free region.** Admit 01 and a continuous positive nonincreasing eta on [T*,infinity), with eta<=1/2 and eta(t)->0, such that beta<=1-eta(|gamma|) for |gamma|>=T*. **(A2) Uniform density.** For fixed A_d>0, B>=1, C>=0, C_d>0 and sigma_0 in (1/2,1), assume N(sigma,T)<=C_d T^[A_d(1-sigma)^B](log T)^C for all sigma_0<=sigma<1 and T>=T*. Here N(sigma,T) counts beta>sigma and 0=T*)[eta(t)u+2log t], P_0(y)=(u/W(y))^(BC) exp(A_d W(y)^(B+1)/u^B). Assume W=o(u) and W/log u->infinity. These imply eventual positivity and P_0>=1. The infimum is not replaced by an asymptotic equivalent inside an exponential. **Lemma.** There are fixed C_S,C_I>0 and y_0 such that for every real y>=y_0, S(y):=sum_rho y^(beta-1)/|rho(rho-1)| <=C_S P_0(y)exp(-W(y)), |I_theta(y)|<=C_I P_0(y)exp(-W(y))/u, I_theta(y)=integral_y^infinity (theta(t)-t)(log t+1)/(t^2 log^2 t) dt. The improper integral exists. Constants and the eventual threshold may depend on the fixed inputs, including their remainder constants and finite low zeros; they do not depend on y. No uniform numerical values across input families are asserted. Also log P_0=o(W). This is a bound on the absolute value of a signed integral. It does not assert the same rate for the integral of the absolute value of theta(t)-t. No sign cancellation hypothesis is assumed. The generic lemma is conditional on its explicit inputs; their ordinary classical/VK instances are unconditional. ## 2. Proof Define theta(t)=sum_(p<=t)log p and psi(t)=sum_(p^k<=t)log p. Let rho=beta+i gamma run over all nontrivial zeros with multiplicity, counting both signs of gamma. Put S(y)=sum_rho y^(beta-1)/|rho(rho-1)|, M_z=sum_rho 1/|rho(rho-1)| < infinity. The total zero count implies convergence of M_z; the finitely many bounded- height zeros have nonzero denominators. Each summand of S is decreasing in y. For q>0 write W_q(y)=inf_(t>=T*)[eta(t)u+q log t], with u=log y. ### 2.1. A split uniform in the moving parameter For sufficiently small 0=sigma_0 and the density exponent at sigma_r equals r/2. Stieltjes integration, counting both signs of gamma, gives sum_{beta>sigma_r, |gamma|>=T*} |gamma|^(-r) <= 2r integral_{T*}^infinity N(sigma_r,t)t^(-r-1)dt <= 2C_d r integral_1^infinity t^(-1-r/2)(log t)^C dt = 2^(C+2) C_d Gamma(C+1) r^(-C). (U1) The lower boundary contribution is nonpositive and can be dropped; the upper boundary vanishes since N(sigma_r,T)T^(-r)=O(T^(-r/2)(log T)^C). Endpoint atoms can instead be handled with left limits, with the same bound. The gamma integral is valid for every real C>=0, not just integer C. The constant in this estimate is independent of r. Write K1=2^(C+2) C_d Gamma(C+1). Finitely many zeros below T* lie to the left of sigma_r for all sufficiently small r. For every zero with beta<=sigma_r, including these finite low zeros, sum y^(beta-1)/|rho(rho-1)| <= M_z exp(-delta u), M_z=sum_rho 1/|rho(rho-1)|=gamma^2, and the zero-free region gives y^(beta-1)/|rho(rho-1)| <=exp(-[(1-beta)u+(2-r)log|gamma|]) |gamma|^(-r) <=exp(-W_{2-r}(y)) |gamma|^(-r). Sum this inequality using (U1). It proves S(y) <= M_z exp(-delta u)+K1 r^(-C)exp(-W_{2-r}(y)). (U2) There is no substitution into a theorem with uncontrolled constants: (U2) is proved uniformly first. **Endpoint substitution.** Set delta=W/u, r=2A_d(W/u)^B. Since W=o(u), all preceding small-r conditions hold eventually. Moreover, pointwise comparison inside the infimum gives W_{2-r}(y) >= (1-r/2) W(y). Indeed eta(t)u+(2-r)log t is at least (1-r/2) times eta(t)u+2log t. Substitute this and delta u=W into (U2): S(y) << P_0(y) exp(-W(y)), (U3) P_0(y)=(u/W(y))^(BC) exp(A_d W(y)^(B+1)/u^B). The fixed factor (2A_d)^(-C) is absorbed into the constant. Eventually u/W>1, so P_0>=1 and also absorbs the first term of (U2). In particular log P_0=BC log(u/W)+A_d W^(B+1)/u^B=o(W). The result reaches the endpoint with a specified secondary loss. It does not in general show S=O(exp(-W)). ### 2.2. The primitive formula and the integrated tail Use the primitive explicit-formula input (A3), normalized as F(x)=integral_2^x (psi(t)-t)dt =-sum_rho x^(rho+1)/(rho(rho+1))-x log(2pi)+O(log x). The zero series is absolutely and uniformly convergent on compact intervals: the numerators are bounded there and the reciprocal quadratic denominators are summable. On a finite interval [y,T], integration by parts followed by termwise integration therefore gives integral_y^T (psi(t)-t)/t^2 dt =sum_rho [y^(rho-1)-T^(rho-1)]/[rho(rho-1)]+O(1/y). The error is uniform in T>=y>=2. The linear term contributes O(1/y), and the primitive O(log x) remainder contributes O(log y/y^2). Since beta<1, the T-dependent zero sum tends to zero by dominated convergence, dominated by the summable reciprocal denominators. Thus the improper integral exists and J_psi(y)=integral_y^infinity (psi(t)-t)/t^2 dt =sum_rho y^(rho-1)/(rho(rho-1))+O(1/y), |J_psi(y)|<=S(y)+C_J/y. This avoids interchanging an unsmoothed zero series with an infinite integral. It also requires no reciprocal factor (1-beta)^(-1). ### 2.3. The weighted theta tail and the prime-power correction Let Delta(t)=theta(t)-t, and define I_psi and I_theta by integrating, respectively, psi(t)-t and Delta(t) against a(t)/t^2 below. Their existence will follow from the following argument. The bound S(y)+C_J/y from Section 2.2 is a decreasing exact envelope: each zero has Re(rho)<1 and the nonnegative series converges. Set a(t)=1/log t+1/log^2 t. Integration by parts first on a finite interval, then at infinity, gives I_psi(y)=a(y)J_psi(y)+integral_y^infinity a'(t)J_psi(t) dt, |I_psi(y)|<=2a(y)(S(y)+C_J/y). (I8) The endpoint a(T)J_psi(T) vanishes, and the integral against |a'| converges by the decreasing envelope and integral_y^infinity |a'|=a(y). This establishes existence of the weighted improper integral without requiring the majorant P_0(t)exp(-W(t)) itself to be decreasing. Let J_theta be the integral defining J_psi with psi replaced by theta. Its convergence follows from the first correction below. Keep the two prime-power corrections distinct: Q_J(y)=J_psi(y)-J_theta(y) =integral_y^infinity (psi(t)-theta(t))/t^2 dt >=0, Q_I(y)=I_psi(y)-I_theta(y) =integral_y^infinity (psi(t)-theta(t))a(t)/t^2 dt >=0. There are at most log(t)/log(2) exponents k>=2 in psi-theta. Each inner sum over p<=t^(1/k) is at most sqrt(t)log t, by counting integers instead of primes. Hence psi(t)-theta(t)=O(sqrt(t)log^2 t). This yields a convergent Q_J=O(log^2 y/sqrt y), and, using a(t)<=2/log t for t>=4, Q_I=O(log y/sqrt y), I_theta=I_psi-Q_I. (I9) For example, integral_y^infinity (log t)t^(-3/2)dt =(2log y+4)/sqrt y. This proves existence of I_theta and controls its absolute value by |I_psi|+Q_I. No O(1/y) estimate for Q_J is used. A sharper Theta(1/sqrt y) statement for Q_J is separate and unnecessary here. Now set A(y)=P_0(y)exp(-W(y))/u. Then (log y/sqrt y)/A(y)=u^2 exp(W-u/2)/P_0 -> 0, (1/(yu))/A(y)=exp(W-u)/P_0 -> 0. Together with (I8), these yield |I_theta|<=C_I A eventually for a fixed C_I after absorbing the prime-power and explicit-formula remainders. The fixed C_I includes these analytic remainders. The later choice of K_0 will leave only the two arithmetic corrections in Section 4 to absorb. This proves the lemma. To make the secondary-factor limit explicit, BC log(u/W)=O(log u)=o(W), while A_d W^(B+1)/u^B=A_d W(W/u)^B=o(W). ## 3. Standard instances and interface to counting For eta(t)=a/log t above a fixed cutoff, elementary minimization eventually gives W=2sqrt(2au). With B=1,C=0, P_0=exp(8A_d a), a fixed factor. Thus |I_theta(y)|=O(exp(-2sqrt(2a log y))/log y). For eta(exp v)=a v^(-2/3)(log v)^(-1/3) and B=3/2,C=15, the known order W comparable to u^(3/5)/(log u)^(1/5) implies P_0=O((u/W)^(45/2)), |I_theta(y)|=O((u/W)^(45/2)exp(-W)/u). The order estimate only bounds the secondary factor; W remains the exact infimum in exp(-W). Proof of that elementary order comparison is in the aggregation note (internal record: global-aggregation-2026-09-14.md; not publicly available), Section 4. For application, set A(y)=P_0 exp(-W)/u. The lemma supplies the analytic constant C_I. Choosing a fixed K_0>C_I and H=K_0 A is a later step: centered inclusion (internal record: centered-inclusion-2026-09-14.md; not publicly available) must absorb the arithmetic remainders and show actual candidates lie in the counted relaxation. The near-pattern partition count and global aggregation are separate. Neither Mertens normalization, Brun–Titchmarsh nor an ENS partition theorem is needed for the analytic lemma itself. No candidate cardinality conclusion is made here. ## 4. Imported sources and attribution The density framework and the Jutila (B=1,C=0) and Ivić (B=3/2,C=15) instances are explicitly recorded by [Johnston, (2.2) and the discussion after (2.7)/(2.9)](https://arxiv.org/html/2411.13791v1). Their original complete proofs remain imports. Johnston's pointwise PNT conclusion and its minimizer assumptions are not used as a theorem about this tail. The primitive formula is located in [Kumchev's lecture notes, Chapter 2, Exercise 16, printed p.38](https://tigerweb.towson.edu/akumchev/distributionofprimesnotes.pdf). It is stated for psi_1(x)=sum_(n<=x)(x-n)Lambda(n); the conversion F=psi_1-x^2/2+2 and zeta'(0)/zeta(0)=log(2pi) give (A3). This is a classical identity stated in a source, not a new proof of that identity. Concrete zero-free-region witnesses are available in [Mossinghoff–Trudgian–Yang, Theorems 1.1 and 1.3](https://arxiv.org/pdf/2212.06867). No claim of optimized constants is needed for this lemma. [Zhao, Section 2 and Lemma 11](https://arxiv.org/html/2411.18903v2) provide related integrated-error expressions and distinct conditional/sign or squared-error results. The inspected statements are not imported as the present unconditional W_2/P_0 bound. This note derives that bound from (A1)–(A4). A derivation in this project is not evidence of priority; an equivalent estimate or a short adaptation elsewhere remains a literature question. ## 5. Evidence and questions for a specialist The proof body is extracted from manuscript Section 3, retaining the uniform split, finite-interval integration and two distinct prime-power corrections. The source chain was independently reviewed as recorded in the global review response (internal record: global-route-review-response-2026-09-14.md; not publicly available). The standalone audit (internal record: spectral-tail-standalone-audit-2026-09-15.md; not publicly available) independently reviewed this presentation at commit bfb3b83. The maintenance response (internal record: spectral-tail-audit-response-2026-09-15.md; not publicly available) records its coverage, a report-only inequality correction, and the presentation edits made after the reviewed snapshot. It is not a Lean theorem and introduces no stronger conclusion than the existing analytic chain. Current source-comparison qualifications are in the recovered-source response (internal record: recovered-sources-review-response-2026-09-14.md; not publicly available). A focused specialist review can answer: 1. Does an existing weighted explicit-formula or density theorem already state this exact absolute signed-tail bound, including P_0 and uniformity as the density split approaches beta=1? If so, give the theorem and parameter map. 2. Do the imported density ranges and primitive remainder justify every uniform constant and limit passage here? Keep finite low zeros and both signs of ordinates explicit. 3. Is the secondary factor an avoidable loss of this split? A stronger estimate requires its own proof; it is not assumed in the counting manuscript. The user-run Claude audit found no mathematical defect in the deduction from (A1)–(A4). It did not re-access Kumchev, re-prove the imported analytic theorems, or re-derive the VK order comparison in this pass. Specialist attribution review and the separate remaining counting-source coverage remain open.