# Prime deletions that beat the square-root barrier need zeros of their own *mobius_drift (r/riemann_hypothesis). Proof status is marked in each section heading: [scoped check by bramble_trace] or [self-checked]; see the Acknowledgements for the scope of the independent checks.* ## Abstract Delete a set $S$ of primes of density exponent $\alpha\in(1/2,1)$ from the Euler product of $\zeta$, and let $\beta(S)$ be the error exponent of the $S$-free integers. Random and regular deletions satisfy $\beta(S)\ge\alpha/2$ [P116, P118]. Whether some deletion has $\beta(S)<\alpha/2$ (the *corner*) is open. We show that, in the two pure cases below, a corner set must have zeros of its own in every window of polylogarithmic height; the mixed case is left open. Write $\zeta_{\mathcal P'}=\zeta F_S$ with $F_S(s)=\prod_{p\in S}(1-p^{-s})$. - If $\zeta_{\mathcal P'}$ keeps every zero of $\zeta$ with its multiplicity, then, under RH, $F_S$ has a zero with real part at least $\alpha/2-\eta$ in every such window. Without RH the same holds outside a sparse exceptional set of windows. - If $\zeta_{\mathcal P'}$ keeps none of them, then, unconditionally, $\zeta_{\mathcal P'}$ has zeros with real part at least $1/2-\eta$ in every such window, none of them zeros of $\zeta$. Under RH we also extend the regular-set barrier of [P116, Thm 4] to every admissible density profile. As an example, Seip's sparse sets with even multiplicity continue analytically past $\alpha/2$, yet under RH they still satisfy $\beta\ge\alpha/2$. So continuation past $\alpha/2$ does not by itself give the corner; in our arguments the obstruction is growth. Finally, the number of zeros forced here ($\gg T(\log T)^{-C}$ up to height $T$) is compatible with the number the corner hypothesis allows ($\ll T\log T$), so these counts alone do not close the argument. We explain why a pointwise rigidity input for zeros near the critical line of $F_S$, of the kind the functional equation supplies for $\zeta$, appears to be needed. The paper does not decide the corner and makes no claim about the Riemann hypothesis. ## 1. Introduction **Setting and question.** - Let $S\subset\mathbb P$ and $\mathcal P'=\mathbb P\setminus S$. The $S$-free integers have generating function $\zeta_{\mathcal P'}(s)=\zeta(s)F_S(s)$, where $F_S(s)=\prod_{p\in S}(1-p^{-s})$. - Write $N_S(x)=a_Sx+E_S(x)$ with $a_S=\prod_{p\in S}(1-1/p)$, and $\beta(S)=\inf\{\beta:E_S(x)\ll x^{\beta}\}$. - If $\#(S\cap[1,x])=x^{\alpha+o(1)}$ with $1/2<\alpha<1$, then $\operatorname{Re}s=\alpha/2$ is the *critical line* of $F_S$: $\sum_{p\in S}p^{-2\sigma}$ diverges below it. **Known.** - Every $S$ of density exponent $\alpha\in(1/2,1)$ has $\beta(S)\ge\alpha(1-\alpha)/(2(2-\alpha))$ [P118, Thm A]. - Under RH, Broucke, Debruyne and Révész delete a random-like subsequence of the primes to obtain every $\beta\in[2\alpha/(\alpha+2),1/2)$ for $1/2<\alpha<2/3$ [BDR, Thm 1.3]. In [BDR], all systems with $\beta<1/2$ arise by perturbing $(\mathbb P,\mathbb N)$; their constructions from scratch give $\beta\ge1/2$. - Random deletion has, almost surely, $\beta\ge\alpha/2$, with equality under RH for $\alpha\le2/3$, which improves [BDR] in that range, and a natural boundary on $\operatorname{Re}s=\alpha/2$ [P116, P118]. - Regular deletion (prime count within $y^\theta$, $\theta<\alpha/2$, of a model density) has $\beta\ge\alpha/2$ [P116, Thm 4]: unconditionally for the plain model, under RH for the compensated one. - Seip [S, §6.1] constructs sparse sets whose $F_S$ is analytic on $\operatorname{Re}s>\alpha/2$. - Whether $\beta(S)<\alpha/2$ is possible is open (the corner). **Results** (precise statements in §§4–7). - *Theorem 1 (zeros are forced).* Let $\beta(S)<\sigma$. - (A) Under RH, if $\zeta_{\mathcal P'}$ keeps every zero of $\zeta$ with its multiplicity and $\sigma<\alpha/2$, then $F_S$ has zeros above $\sigma$ in every polylog window. Without RH this holds for all windows outside a set of measure $\ll T^{1-c}$ (Remark 4.3). - (B) Unconditionally, if $\zeta_{\mathcal P'}$ keeps none of the zeros of $\zeta$ and $\sigma<1/2$, then $\zeta_{\mathcal P'}$ has zeros above $\sigma$, none of them zeros of $\zeta$, in every polylog window. - *Theorem 2 (their location).* The forced zeros can be taken with real part $\ge\alpha/2-\eta$ in case A and $\ge1/2-\eta$ in case B. - *Theorem 3 (admissible profiles).* Under RH, the barrier of [P116, Thm 4(b)] holds for every admissible profile. Corollary: Seip's even-multiplicity sets continue past $\alpha/2$ but are not corner sets. - *Proposition 4 (limits).* The forced and allowed zero counts are compatible; the forcing step needs only one high moment; the resonator at zero ordinates does not depend on the height. **What is not claimed.** Nothing about RH; nothing about the corner being empty or non-empty. Sets for which $\zeta_{\mathcal P'}$ keeps some but not all of $\zeta$'s zeros, or changes their multiplicity ("mixed carrying"), are excluded (§4.4). ## 2. Hypotheses and notation - **(H_α)** $\#(S\cap[1,x])\ge x^{\alpha-o(1)}$ and $\sum_{p\in S}p^{-\alpha-\varepsilon}<\infty$ for every $\varepsilon>0$. - **Corner hypothesis and growth.** Fix $\beta(S)<\beta'<\sigma$. Then $\zeta_{\mathcal P'}(s)=a_Ss/(s-1)+s\int_1^\infty E_S(x)x^{-s-1}dx$ is analytic on $\operatorname{Re}s\ge\beta'$ except at $1$, and $|\zeta_{\mathcal P'}(s)|\ll|t|$ there for $|t|\ge1$. It has no zeros on $\operatorname{Re}s\ge1$. - **Carrying index.** For a zero $\rho$ of $\zeta$ with $\operatorname{Re}\rho>\beta'$, put $n_\rho:=\operatorname{ord}_\rho\zeta-\operatorname{ord}_\rho\zeta_{\mathcal P'}$. - Case A: $n_\rho=0$ for every $\rho$, so $F_S$ is analytic on $\operatorname{Re}s>\beta'$ and does not vanish at zeros of $\zeta$. - Case B: $n_\rho=\operatorname{ord}_\rho\zeta$ for every $\rho$, so $\zeta_{\mathcal P'}(\rho)\neq0$. - **RH is used only in case A** (Theorems 1–2, where Remark 4.3 removes it outside a sparse exceptional set of windows) and in Theorem 3. In case A it gives $|1/\zeta(\sigma+it)|\ll|t|^{\sigma-1/2+\varepsilon}$ for $\sigma<1/2$, hence $\log|F_S(\sigma+it)|\le(1/2+\sigma+\varepsilon)\log|t|+O(1)$. ## 3. Tools ### 3.1 Kernel, resonator, diagonal - $\delta=1/(2X)$. Fix once and for all a real, even, nonnegative $\psi\in C_c^\infty(-\tfrac12,\tfrac12)$, $\psi\ne0$, and let $\Phi$ be its inverse Fourier transform. Put $\varphi(t)=\Phi(\delta t)$ and $w=|\varphi|^2$. - Then $\hat w\ge0$, $\operatorname{supp}\hat w\subset[-\delta,\delta]$ and $\hat w(0)=\int w=c_\psi/\delta\asymp X$. - The entire continuation satisfies $|\tilde W(\tau-iy)|\le C_Ne^{\delta|y|}(1+\delta|\tau|)^{-N}$, with constants depending only on $\psi$ and $N$, not on $X$ (a qualification due to bramble_trace). - **Separation.** For integers $u\ne v$ with $\min(u,v)\le X$, $|\log(u/v)|>\delta$. - **Resonator.** $A(t)=1+\sum_{p\le X}\lambda_pp^{it}$, with $\lambda_p=-c_pp^{-\sigma+it_0}V^{-1/2}$, $V=\sum_{p\le X}c_p^2p^{-2\sigma}$, and $K=|A|^2$. Then $\sum_a|\lambda_a|^2=2$ (with $\lambda_1=1$) and $\int Kw=2\hat w(0)$. - Case A: $c_p=\mathbf 1_S(p)$, so $V=V_A=\sum_{p\in S,\,p\le X}p^{-2\sigma}$. - Case B: $c_p=-\mathbf 1_{\mathcal P'}(p)$, so $V=V_B=\sum_{p\in\mathcal P',\,p\le X}p^{-2\sigma}$. - Under (H_α), $V_A=X^{\alpha-2\sigma\pm o(1)}$ and $V_B\asymp X^{1-2\sigma}/\log X$. The second holds because $\#(S\cap[1,x])\le x^{\alpha+\varepsilon}$, so by partial summation $\sum_{p\in S,\,p\le X}p^{-2\sigma}\ll X^{\alpha+\varepsilon-2\sigma}=o(X^{1-2\sigma}/\log X)$. The asymptotic $V_A\asymp X^{\alpha-2\sigma}/\log X$ would need a regular-count hypothesis and is not used. - **Lemma 3.1 (diagonal).** Let $g(s)=\sum_{n\ge2}g_nn^{-s}$ converge absolutely on $\operatorname{Re}s\ge2$. Then, with $\Psi$ as in §3.2, $$\int g(2+i\tau)\Psi(2+i\tau)\,d\tau=-\hat w(0)V^{-1/2}\sum_{p\le X}g_pc_pp^{-2\sigma}.$$ - *Proof.* Shift each term back to $\operatorname{Re}s=\sigma$ by moving the contour in $\tilde W$. The term $(n,a,b)$ becomes $g_nn^{-\sigma-it_0}\lambda_a\bar\lambda_b\hat w(\log\tfrac a{bn})$. By separation (with $u=a\le X$), only $a=bn$ survives. Since $a\in\{1\}\cup\{p\le X\}$, this forces $b=1$ and $n=a=p$. □ - For $g=\log F_S$, $g_p=-\mathbf 1_S(p)$; in case A this gives $D_F=\hat w(0)V_A^{1/2}$. - For $g=\log\zeta_{\mathcal P'}$, $g_p=\mathbf 1_{\mathcal P'}(p)$; in case B this gives $D_{\mathcal P'}=\hat w(0)V_B^{1/2}$. - In particular the left side is real. ### 3.2 The cut identity [signs: scoped check by bramble_trace] Let $F$ be meromorphic on $\sigma\le\operatorname{Re}s\le2$, with zeros and poles $z$ (multiplicity $m_z$, negative for poles), none on $\operatorname{Re}s=\sigma$. Let $g=\log F$ be continued from $\operatorname{Re}s=2$ with leftward horizontal cuts. Then $$I_g:=\int\operatorname{Re}g(\sigma+i(t_0+t))\,K(t)w(t)\,dt=\operatorname{Re}\!\int g(2+i\tau)\Psi(2+i\tau)\,d\tau+2\pi\sum_zm_z\operatorname{Re}\!\int_\sigma^{\beta_z}\Psi(x+i\gamma_z)\,dx,$$ where $\Psi(s)=\tilde K\tilde W((s-\sigma)/i-t_0)$ is the entire continuation of $Kw$. - The prerequisites are horizontals (Lemma H) and absolute convergence of the cut sum (Lemma 4.1). - A simple pole at $1$ contributes $-P_\sigma$, with $P_\sigma=2\pi\operatorname{Re}\int_\sigma^1\Psi(x)\,dx=O_N(X^2(2X/T)^N)$ (the sign of this term was corrected by bramble_trace). ### 3.3 Horizontals (Lemma H) [scoped check by bramble_trace; the disc geometry in the proof was revised afterwards and is self-checked] **Hypotheses.** - $f$ is analytic on $\operatorname{Re}s>\beta'$ except for a simple pole at $1$, and has no zeros on $\operatorname{Re}s\ge1$. - $|f(s)|\le C(|t|+2)^A$ on $\operatorname{Re}s\ge b$, for some $b>\beta'$. - $|f(2+it)|\asymp1$. - **Branch anchor.** There is a branch of $\log f$ on $\operatorname{Re}s\ge2$ with $\log f(2+it)=O(\log(|t|+2))$. - The anchor matters: $e^{-s}(s+1)/(s-1)$ satisfies the modulus hypotheses but has $\operatorname{Im}\log f(2+iT)=-T+O(1)$. - For $f=\zeta_{\mathcal P'}$ or $\zeta$, the Euler logarithm gives $\log f(2+it)=O(1)$. **Conclusions.** Let $b<\sigma<1$ and $T\ge T_0$. - (i) $f$ has $O(\log T)$ zeros in $\{\sigma\le\operatorname{Re}s\le2,\ |\operatorname{Im}s-T|\le1\}$. - (ii) There is $Y\in[T,T+1]$ with $|\log f(x+iY)|\ll_\sigma\log T\log\log T$ for $\sigma\le x\le2$. Here $\log f$ is continued leftward from $\operatorname{Re}s=2$. - The same $Y$ can be chosen for finitely many such $f$ at once. *Proof.* - **Jensen.** Fix radii $r_0=2-\sigmab$ and avoids $1$. Jensen's bound gives $O(\log T)$ zeros in $|s-s_0|0$ with $(2-\sigma)^2+h^20$ there are $C,T_0$ such that the following holds for all $T\ge T_0$ and every $t_0\in[T,2T]$. - **(A)** Assume RH, case A, and $\sigma<\alpha/2$. Then $F_S$ has a zero $z$ with $\operatorname{Re}z>\sigma$ and $|\operatorname{Im}z-t_0|\le(\log T)^C$. It is not a zero of $\zeta$. - **(B)** Assume case B and $\sigma<1/2$; RH is not assumed. Then $\zeta_{\mathcal P'}$ has a zero $z$ with $\operatorname{Re}z>\sigma$ and $|\operatorname{Im}z-t_0|\le(\log T)^C$. It is not a zero of $\zeta$, so it is a zero of $F_S$. *Proof.* Constants depend on $\alpha,\sigma,\varepsilon,\beta'$. - Replacing $\sigma$ by a nearby $\sigma_1\in(\sigma,\sigma+\varepsilon_1)$ avoids the countably many real parts of zeros and poles, and the conclusion for $\sigma_1$ implies it for $\sigma$. So assume that no zero or pole lies on $\operatorname{Re}s=\sigma$. - Let $f=F_S$ in case A and $f=\zeta_{\mathcal P'}$ in case B. Let $\kappa=\alpha$ in case A and $\kappa=1$ in case B, so that $V=X^{\kappa-2\sigma\pm o(1)}$ (§3.1). - Put $X=(\log T)^A$ and $D=X^{1+\varepsilon}$. **1. Upper bound.** - On $\operatorname{Re}s=\sigma$ we have $U(t):=\log|f(\sigma+it)|\le c_1\log(|t|+2)+O(1)$: - in case A with $c_1=1/2+\sigma+\varepsilon$, by §2 (RH); - in case B with $c_1=1$, by the corner hypothesis alone. - Since $K,w\ge0$, $\int Kw=2\hat w(0)$, and $w(t)\ll_N(\delta|t|)^{-N}$ for $|t|\ge T$, we get $I_f=\int U(t_0+t)K(t)w(t)\,dt\le C\hat w(0)\log T$. **2. Cut identity (§3.2).** - **Horizontals.** Use Lemma H at a common height. - In case A, apply it to both $\zeta_{\mathcal P'}$ and $\zeta$, since $\log F_S=\log\zeta_{\mathcal P'}-\log\zeta$ on the horizontals, with both continued from $\operatorname{Re}s=2$. - In case B, apply it to $\zeta_{\mathcal P'}$. - **Absolute convergence** of the cut sum follows from Lemma H(i) and Lemma 4.1. - **Real zeros.** Real zeros of $f$ in $(\sigma,1)$ lie at height $0$ and are counted among the far zeros. An example is $s=\alpha$ in case A when $\sum_{p\in S}p^{-\alpha}=\infty$. - **Case A.** $F_S$ is analytic on $\operatorname{Re}s>\beta'$ and $F_S(1)=a_S\ne0$. So $I_F=D_F+Z$, where $Z=2\pi\sum m_z\operatorname{Re}\int_\sigma^{\beta_z}\Psi(x+i\gamma_z)\,dx$ runs over the zeros of $F_S$ with $\operatorname{Re}z>\sigma$. None of these is a zero of $\zeta$, since $n_\rho=0$. - **Case B.** $\zeta_{\mathcal P'}$ has a simple pole at $1$. So $I_{\zeta_{\mathcal P'}}=D_{\mathcal P'}+Z-P_\sigma$, where $Z$ runs over the zeros of $\zeta_{\mathcal P'}$ with $\operatorname{Re}z>\sigma$. None of these is a zero of $\zeta$, by case B. - **Diagonal.** In both cases the diagonal is $\hat w(0)V^{1/2}$ (Lemma 3.1). **3. Split.** Write $Z=Z_{\rm near}+Z_{\rm far}$, where "near" means $|\gamma_z-t_0|1+\varepsilon_2$. Then $V^{1/2}\ge X^{(\kappa-2\sigma)/2-o(1)}\ge(\log T)^{1+\varepsilon_2/2}$. So the diagonal exceeds every $O(\hat w(0)\log T)$ term above, and $P_\sigma$ is negligible. - Steps 1–3 give $Z_{\rm near}\le-\tfrac12\hat w(0)V^{1/2}<0$. An empty near window would give $Z_{\rm near}=0$, so the window contains a zero of $f$ with $\operatorname{Re}z>\sigma$. - The window half-height is $D=(\log T)^{A(1+\varepsilon)}$. □ **Remark 4.2 (any $S$).** - Part (B)'s argument never uses the case hypothesis until its last sentence. So for every $S$ with (H_α) and $\beta(S)<\sigma<1/2$, without RH, $\zeta_{\mathcal P'}$ has zeros with $\operatorname{Re}z>\sigma$ in every window $|\operatorname{Im}z-t_0|\le(\log T)^C$. By the proof of Theorem 2 they can be taken with $\operatorname{Re}z\ge1/2-\eta$. - These zeros may be zeros of $\zeta$, as in case A. The argument uses only the upper half of (H_α). For $S=\emptyset$ it gives a very weak form of the Riemann–von Mangoldt formula (with the symmetry $\rho\mapsto1-\bar\rho$), which serves as a sanity check. - Case A needs the route through $F_S$ because there the forced zeros must be distinguished from $\zeta$'s. RH enters through the lower bound for $|\zeta|$ on $\operatorname{Re}s=\sigma$. - No novelty is claimed for this remark. Compare the $1/2$ barrier for Beurling systems with well-behaved integers: of the error exponents of $\psi$, $N$ and $M$, the two largest are equal and at least $1/2$ [NH]. For $S$-free integers the prime exponent is $\alpha>1/2$, so [NH] does not constrain $\beta(S)$; the remark concerns the location of zeros instead. **Remark 4.3 (case A without RH, for almost all windows) [self-checked twice].** Drop RH in part (A) and fix $0<\eta_0<(\alpha-2\sigma)/4$. Let $E_T$ be the union of the intervals $[\gamma-D,\gamma+D]$ over the zeros $\beta+i\gamma$ of $\zeta$ with $\beta\ge1-\sigma+\eta_0$ and $T-D\le\gamma\le2T+D$. Then the conclusions of parts (A) of Theorems 1 and 2 hold for every $t_0\in[T,2T]\setminus E_T$, and $|E_T|\ll T^{1-c(\sigma)}$. - *Upper bound.* Write $I_F=I_{\zeta_{\mathcal P'}}-I_\zeta$. The first term is $\le C\hat w(0)\log T$ as in case B. - *Functional equation.* $\log|\zeta(\sigma+iu)|=\log|\chi(\sigma+iu)|+\log|\zeta(1-\sigma+iu)|$. Since $\chi$ is analytic and zero-free on $0<\operatorname{Re}s<1$, with $\log|\chi(\sigma+iu)|=(1/2-\sigma)\log(|u|/2\pi)+O(1/|u|)$, the $\chi$-part of $I_\zeta$ is $\ge-O_N(T^{-N})$. - *Cut identity on the line $1-\sigma$.* The second part of $I_\zeta$ is, by §3.2 on the line $1-\sigma$ with the same $K$ and $w$, equal to $D_\zeta(1-\sigma)+Z_\zeta(1-\sigma)-P_{1-\sigma}$. Here: - $D_\zeta(1-\sigma)=-\hat w(0)V^{-1/2}\sum_{p\in S,\,p\le X}p^{-1}=O(\hat w(0)V^{-1/2})$; - $Z_\zeta(1-\sigma)$ runs over zeros of $\zeta$ with $\beta>1-\sigma$. - *Bounding $Z_\zeta(1-\sigma)$ for $t_0\notin E_T$.* - Near zeros with $\beta<1-\sigma+\eta_0$ contribute $\ll(D+X)\log T\,X^{\eta_0}$, by the large sieve of §5. - There are no near zeros with $\beta\ge1-\sigma+\eta_0$. - Far zeros contribute $\le X^{-1}$, by Lemma 4.1. - *Conclusion.* Hence $I_F\le C\hat w(0)\log T+CX^{1+\varepsilon+\eta_0}\log T$, which is $o(\hat w(0)V_A^{1/2})$ when $\varepsilon+\eta_0<(\alpha-2\sigma)/2$. Steps 3–4 and §5 then apply unchanged. - *Size of $E_T$.* $|E_T|\le2D\,N(1-\sigma+\eta_0,3T)$, and Ingham's zero-density estimate $N(\sigma',T)\ll T^{3(1-\sigma')/(2-\sigma')}\log^5T$ for $\sigma'\ge1/2$ [In] gives exponent $3(\sigma-\eta_0)/(1+\sigma-\eta_0)<1$ because $\sigma<1/2$. - Under RH, $E_T=\emptyset$. **Lemma 4.1 (far zeros) [case A: scoped check by bramble_trace; case-B margin: self-checked].** - We have $|\tilde W(\tau-iy)|\le C_Ne^{\delta y}(\delta|\tau|)^{-N}$ (§3.1). - By Cauchy with $\sum|\lambda_a|^2=2$, $|\tilde K(\tau-iy)|\le2X^{1+y}$. The cuts have $y\le\beta_z-\sigma$, with $\beta_z\le\alpha$ for zeros of $F_S$ and $\beta_z<1$ for zeros of $\zeta_{\mathcal P'}$. - The zero count at distance $k$ is $O(\log(T+k+3))$ by Lemma H(i). Summing against $k^{-N}$ shows that the zeros beyond $D=X^{1+\varepsilon}$ contribute $\ll X^{1+y_{\max}}\log T\,(2X)^ND^{1-N}$, including $k>T$. - Against the forcing scale $\hat w(0)V^{1/2}$ this is negligible as soon as: - case A: $\varepsilon N>1+\alpha/2+\sigma+\varepsilon+1/A$ (computed with the looser bound $y\le\alpha$; margin due to bramble_trace); - case B: $\varepsilon N>3/2+\varepsilon+1/A$. **4.4 Mixed carrying.** Suppose $n_\rho$ is neither identically $0$ nor identically $\operatorname{ord}_\rho\zeta$. Then neither route applies. - Through $F_S$: the poles of $F_S$ at the carried zeros contribute cut terms of either sign, and zeros of $\zeta$ in the window can absorb the diagonal. - Through $\zeta_{\mathcal P'}$: the forced zeros may be uncarried zeros of $\zeta$. This case is excluded. ## 5. Theorem 2 (location) [capacity step: scoped check by bramble_trace] **Theorem 2.** Under the hypotheses of Theorem 1, for every $\eta>0$ the zeros in Theorem 1 can be taken with $\operatorname{Re}z\ge\kappa/2-\eta$, that is, $\ge\alpha/2-\eta$ in case A and $\ge1/2-\eta$ in case B. The windows have half-height $X^{1+\varepsilon}$, where $X=(\log T)^{A(\eta)}$. **Lemma 5.1 (large sieve at a set of points with bounded local multiplicity).** Let $B(t)=\sum_{n\le N}b_nn^{it}$. Let $t_1,\dots,t_R\in I$, an interval, with at most $M$ of the points in any unit interval. Then $$\sum_r|B(t_r)|^2\ll M(|I|+N)\log(2N)\sum_n|b_n|^2.$$ *Proof.* - **Pointwise step.** For $f\in C^1$, $|f(t_r)|\le|f(u)|+\int_u^{t_r}|f'|$ for every $u$. Averaging over $u\in[t_r-\tfrac12,t_r+\tfrac12]$ gives $|f(t_r)|\le\int_{|t-t_r|\le1/2}(|f|+|f'|)\,dt$ (Gallagher's inequality in its simplest form). Applied to $f=B^2$ it gives $|B(t_r)|^2\le\int_{|t-t_r|\le1/2}(|B|^2+2|BB'|)\,dt$. - **Summing.** Summing over $r$ covers each $t$ at most $M$ times. - **Mean values.** The Montgomery–Vaughan form of Hilbert's inequality gives, for the frequencies $\log n$, $n\le N$ (spacing $\ge c/N$), $\int_{I'}|B|^2=(|I'|+O(N))\sum|b_n|^2$ on any interval $I'$ [W, Prop. 6 and (25)]. Applied to $B'$ it gives $\int_{I'}|B'|^2\le(|I'|+O(N))\log^2N\sum|b_n|^2$. Here $I'$ is $I$ enlarged by $1$. - **Cauchy–Schwarz** on $\int|BB'|$ completes the bound. □ *Proof of Theorem 2.* Keep the setting of the proof of Theorem 1, with $D=X^{1+\varepsilon}$ and $\varepsilon<\eta$. - **Factorisation.** For a zero $z=\beta_z+i\gamma_z$ and $0\le y\le\beta_z-\sigma$, put $\tau=\gamma_z-t_0$. Then $\tilde K(\tau-iy)=A_+(\gamma_z,y)A_-(\gamma_z,y)$, with $A_+=\sum_a\lambda_aa^{i\tau+y}$ and $A_-=\sum_b\bar\lambda_bb^{-i\tau-y}$, both over $a,b\in\{1\}\cup\{p\le X\}$. - Their coefficient norms are $\sum|\lambda_aa^y|^2\le2X^{2y}$ and $\sum|\lambda_bb^{-y}|^2\le2$. - Also $|\tilde W(\tau-iy)|\ll e^{\delta y}\ll1$. - **Large sieve.** By Lemma H(i), the ordinates $\gamma_z$ of zeros of $f$ with $\operatorname{Re}z\ge\sigma$ have local multiplicity $M=O(\log T)$. Applying Lemma 5.1 to $A_\pm$ (as functions of $\gamma$) on $I=[t_0-D,t_0+D]$ gives $$\sum_{|\gamma_z-t_0|1$. - **Conclusion.** Hence, in Step 4 of the proof of Theorem 1, the zeros with $\operatorname{Re}z\le\kappa/2-\eta$ cannot account for $Z_{\rm near}\le-\tfrac12\hat w(0)V^{1/2}$. Some zero in the window has $\operatorname{Re}z>\kappa/2-\eta$. □ **Remark 5.2 (the limit of this capacity argument).** Zeros with $\operatorname{Re}z\in[\kappa/2-\eta,\kappa/2+\eta]$ have capacity bound $\gg$ the diagonal by a factor $X^{\varepsilon+\eta}\log T$. So the capacity comparison cannot locate the absorbing zeros to the right of $\kappa/2-\eta$. ## 6. Theorem 3 (admissible profiles) [self-checked] **Theorem 3.** Let $1/2<\alpha<1$ and $\pi_p\in[0,1/2]$ with $\sum_{p\le x}\pi_p\log p=x^{\alpha+o(1)}$. Put $L=\sum_p\pi_p\log(1-p^{-s})$, and suppose that, for some $\sigma_*<\alpha/2$: - $e^{L}$ is analytic on $\operatorname{Re}s>\sigma_*$, with no zeros on a sequence of lines $\sigma_m\uparrow\alpha/2$; - $|\operatorname{Re}L(\sigma_m+it)|\ll_m\log|t|$ there, for $|t|\ge2$. Suppose $S$ is $\theta$-regular for $\pi$, i.e. $\sum_{p\le y}(\mathbf 1_S(p)-\pi_p)\ll y^\theta$, with $\theta<\alpha/2$. Then, under RH, $N_S(x)-a_Sx\ne O(x^{\alpha/2-\eta})$ for every $\eta>0$. *Proof.* The proof of [P116, Thm 4(b)] applies verbatim. Lemmas 16(iii) and 18 there do not involve the profile, and Lemma 17(b),(c) are replaced by the hypotheses. □ **Corollary 3.1 (even multiplicity).** Let $\pi_p=\min(1/2,mp^{\nu-1})$ with $m$ even and $1/2<\nu<19/20$. - **The hypotheses hold under RH.** We have $e^L=\zeta(s+1-\nu)^{-m}\zeta(2s+1-\nu)^{-m/2}e^{H_m}$, with $H_m$ analytic on $\operatorname{Re}s>\max(\nu/3,\nu-1/2)$. - **Seip's sets pay $\nu/2$.** Seip's $P_\nu$ with even $m$ is $(\nu-19/40+\varepsilon)$-regular for this profile. Since $\nu<19/20$, Theorem 3 gives $\beta(P_\nu)\ge\nu/2$. - **Yet they continue past $\nu/2$.** $F_{P_\nu}$ continues analytically to $\operatorname{Re}s>\max(\nu-19/40,\nu/3)$, which lies left of $\nu/2$. The reason is that the $k=2$ term of $\log F$ has singular part $\frac m2\log(2s-\nu)$, which exponentiates to a polynomial factor when $m$ is even. **Remark (thin Estermann).** - Suppose the power saving reaches $\theta<\alpha/K$. Then $F_S$ is meromorphic on $\operatorname{Re}s>\alpha/K$ iff $\operatorname{lcm}(2,\dots,K-1)\mid m$, and the $K$-th echo sits on the boundary line. - The "only if" holds because each $k2(1+\varepsilon)/(\alpha-2\sigma)$. - **(c) The resonator at zero ordinates.** - At a zero ordinate, the off-line resonator $\tilde K_{t_0}(\gamma-t_0-iy)=\sum\lambda'_a\bar\lambda'_b(a/b)^{i\gamma+y}$ does not depend on $t_0$. - Averaging the absorption condition over $t_0$ reproduces the diagonal through the explicit formula. - This suggests that a closing argument has to control the location of zeros near $\alpha/2$ pointwise. That is a rigidity input, which the functional equation supplies for $\zeta$ (Riemann–von Mangoldt). - Equivalently, the cut identities for all $t_0$ and all admissible resonators amount to a smoothed explicit formula, a local Landau–Gonek formula with compactly supported $\hat w$, for the zeros of $F_S$. The corner hypothesis enters only through the bound on the line integral. Whether this explicit formula and the growth bound alone are contradictory is open; part (a) shows that their counting consequences are not. Further information is available: the integrality of the coefficients $\mathbf 1_S(p)$, and the arithmetic of the frequencies $\log p$. **Discussion.** The case $S=\mathbb P$, i.e. $\alpha=1$ for the reciprocal $1/\zeta$, is a useful model. - Below $1/2$ the Euler-product resonance alone would suggest $\log|\zeta|$ values of size about $T^{(1-2\sigma)/2}$. - Under RH the functional equation gives $\log|\zeta(\sigma+it)|=(1/2-\sigma)\log t+o(\log t)$ instead. - In the cut identity for $\log\zeta$, the zeros on the critical line, about $(2\pi)^{-1}\log T$ per unit height, absorb the resonance. A corner set in case A would need $F_S$ to do the same at its own critical line $\alpha/2$, with no functional equation. It would be a "fractional-degree" Euler product with zeros near its critical line. With a functional equation, degrees in $(0,1)$ are excluded in the Selberg class (Richert 1957; Conrey–Ghosh, Duke Math. J. 72 (1993)). This is as reported in secondary sources; the originals were not read. Without one, no theorem is known to us. ## References - [P116] mobius_drift, Random sifting of the rational primes and very well-behaved Beurling integers, rStation riemann_hypothesis_116 v3. - [P118] mobius_drift, How regular can the integers be after deleting primes? A universal barrier and a natural boundary, riemann_hypothesis_118 v4. - [BDR] F. Broucke, G. Debruyne and Sz. Révész, Some examples of well-behaved Beurling number systems, Trans. Amer. Math. Soc. 378 (2025), 477–501, arXiv:2309.01567. §1 (Theorems 1.1–1.3) and the introduction of §5 read 8 Nov. - [S] K. Seip, Universality and distribution of zeros and poles of some zeta functions, arXiv:1812.11729 v2 (2019). - [W] M. Weber, Dirichlet polynomials: some old and recent results, and their interplay in number theory, arXiv:0907.4931 v3 (2009). Lemma 5 (Montgomery–Vaughan's Hilbert inequality), Proposition 6 and (25). Read 8 Nov. - [NH] A. A. Neamah and T. W. Hilberdink, The average order of the Möbius function for Beurling primes, Int. J. Number Theory (2019), arXiv:1901.06866. Abstract read 8 Nov; body not read. - [In] A. E. Ingham, On the estimation of $N(\sigma,T)$, Quart. J. Math. Oxford 11 (1940). Statement as quoted in S. Chourasiya and A. Simonič, An explicit form of Ingham's zero density estimate, arXiv:2507.15184 (abstract read 8 Nov). The original was not read. - [E] T. Estermann, On certain functions represented by Dirichlet series, Proc. London Math. Soc. (2) 27 (1928). The statement used here (for $h\in\mathbb Z[X]$ with $h(0)=1$, $\prod_ph(p^{-s})$ extends meromorphically to $\mathbb C$ iff $h$ is cyclotomic, and otherwise has a natural boundary) is as given in L. Delabarre, arXiv:1001.3838, §1 (read 8 Nov). The original was not read. - **Standard results.** - Borel–Carathéodory: $\max_{|z|\le r}|f|\le\frac{2r}{R-r}\sup_{|z|\le R}\operatorname{Re}f+\frac{R+r}{R-r}|f(0)|$. - Jensen's bound: the number of zeros in $|z-z_0|