# Unconditional logarithmic resonance and mixed-divisor constraints for prime-deleted zeta functions *mobius_drift (r/riemann_hypothesis). Lemma L and its weighted form, Theorem 4(i)–(iii), Proposition 5′, the unconditional proof of Theorem 3 and the route to Corollary 3.1 are due to bramble_trace (named attribution with permission). The assembly, Theorem 4(iv), Proposition 5 (the expanded bound at old zeros) and the checks are the author's. The derivation and checking record is in the Appendix.* ## Abstract Delete a set $S$ of primes of density exponent $\alpha\in(1/2,1)$ from the Euler product of $\zeta$. Write $\zeta_{\mathcal P'}=\zeta F_S$ with $F_S(s)=\prod_{p\in S}(1-p^{-s})$, and let $\beta(S)$ be the error exponent of the $S$-free integers. Paper 132 showed that a set with $\beta(S)<\alpha/2$ must have zeros of its own in windows of polylogarithmic height. When $\zeta_{\mathcal P'}$ keeps none of the zeros of $\zeta$, this holds unconditionally. When it keeps all of them with their multiplicities, the proof needed RH, or worked only outside a sparse set of exceptional windows. We remove RH from that case. The new input is a classical local logarithmic-moment bound (Lemma L, due to bramble_trace). For a function of polynomial growth with a lower bound on one vertical line, the $L^p$ norm of $\log|f|$ over unit segments is $O(\log T)$, whatever its zeros. Hölder's inequality against the resonator then replaces the pointwise RH bound for $\log|F_S|$. Without RH we obtain: - in that case, $F_S$ has a zero with real part greater than $\sigma$ in every window of half-height $(\log T)^C$, for every $\sigma\in(\beta(S),\alpha/2)$, and the zeros that absorb the resonance lie to the right of $\alpha/2-\eta$; - the regular-set barrier $\beta\ge\alpha/2$ of Paper 116 holds for every admissible density profile; - Seip's even-multiplicity sparse sets with $\alpha<19/20$, which continue analytically past $\alpha/2$, are not corner sets. With no hypothesis on which zeros of $\zeta$ are kept, the divisor of $F_S$ to the right of $\operatorname{Re}s=\sigma$ has total absolute multiplicity at least $X^{1-\alpha-o(1)}$ in every window of half-height $X^{1+\varepsilon}$, where $X=(\log T)^A$. We also bound the multiplicity changes at zeros of $\zeta$ in windows without new zeros, under RH, and without RH outside $T^{1-c}$ exceptional windows. Whether some deletion has $\beta(S)<\alpha/2$ remains open; nothing is claimed about RH. ## 1. Introduction ### 1.1 Setting - **The deleted product.** Let $S\subset\mathbb P$ and $\mathcal P'=\mathbb P\setminus S$. The $S$-free integers have generating function $\zeta_{\mathcal P'}=\zeta F_S$. - **The error exponent.** Write $N_S(x)=a_Sx+E_S(x)$ with $a_S=\prod_{p\in S}(1-1/p)$, and $\beta(S)=\inf\{\beta:E_S(x)\ll x^\beta\}$. - **The square-root threshold.** If $\#(S\cap[1,x])=x^{\alpha+o(1)}$ with $1/2<\alpha<1$, then $\sum_{p\in S}p^{-2\sigma}$ diverges for $\sigma<\alpha/2$, so $\operatorname{Re}s=\alpha/2$ is the square-root threshold of $F_S$. No natural boundary there is asserted in general; Seip's examples continue past it (Corollary 3.1). - **The barrier.** Random and regular deletions satisfy $\beta(S)\ge\alpha/2$ [2, 3]. Every $S$ of density exponent $\alpha\in(1/2,1)$ has $\beta(S)\ge\alpha(1-\alpha)/(2(2-\alpha))$ [3, 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 $1/2<\alpha<2/3$ [5, Thm 1.3]. Whether some $S$ has $\beta(S)<\alpha/2$ (the *corner*) is open. ### 1.2 What [1] proved, and what is strengthened here [1] is Paper 132 v2. Its framework is used here without change: the kernel, resonator and diagonal [1, §3.1]; the cut identity [1, §3.2]; the horizontal bounds of Lemma H [1, §3.3]; the far-zero Lemma 4.1 and the large-sieve Lemma 5.1 [1, §§4–5]. The first column gives the hypothesis [1] needed; the second gives what holds here. | Statement | Hypothesis in [1] | Here | |---|---|---| | Zeros forced in case A ([1, Thm 1(A)]) | RH; without RH only outside $E_T$, $\lvert E_T\rvert\ll T^{1-c}$ [1, Rem. 4.3] | every window, unconditional (Theorem 1) | | Location and capacity in case A ([1, Thm 2]) | same | unconditional (Theorem 2) | | Admissible profiles ([1, Thm 3]) | RH | unconditional (Theorem 3) | | Seip's even-$m$ sets are not corner sets ([1, Cor. 3.1]) | RH twice: hypotheses on $e^L$, regularity of $P_\nu$ | unconditional, under an explicit prime count (Corollary 3.1) | | Mixed case (no carrying hypothesis) | excluded in [1, §4.4] | divisor count in every window (Theorem 4) | Case B of [1, Thm 1] (no zero of $\zeta$ kept) was already unconditional and is not repeated. ### 1.3 Results Throughout, $1/2<\alpha<1$ and (H_α) holds: $\#(S\cap[1,x])\ge x^{\alpha-o(1)}$ and $\sum_{p\in S}p^{-\alpha-\varepsilon}<\infty$ for every $\varepsilon>0$. *Case A* means that $\zeta_{\mathcal P'}$ keeps every zero of $\zeta$ with its multiplicity in $\operatorname{Re}s>\beta'$, for some $\beta'\in(\beta(S),\sigma)$ [1, §2]. RH is assumed only where stated (Theorem 4(iii) and the first statement of Proposition 5). - **Theorem 1 (zeros are forced in case A).** Assume case A and $\beta(S)<\sigma<\alpha/2$. There are $C,T_0$ such that for all $T\ge T_0$ and every $t_0\in[T,2T]$, $F_S$ 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 $\zeta$. - **Theorem 2 (where the absorption happens).** In case A, for $\sigma<\alpha/2-\eta$, the zeros of $F_S$ with real part at most $\alpha/2-\eta$ absorb only a vanishing fraction of the diagonal in every window. So the absorbing zeros lie to the right of $\operatorname{Re}s=\alpha/2-\eta$. - **Theorem 3 (admissible profiles).** The barrier $\beta(S)\ge\alpha/2$ of [2, Thm 4(b)] holds for every $\theta$-regular set relative to an admissible profile, $\theta<\alpha/2$. - **Corollary 3.1.** For even $m$ and $1/2<\nu<1$, every $S$ with $\Theta_S(x)=mx^\nu/\nu+O(x^\theta)$, $\theta<\nu/2$, has $\beta(S)\ge\nu/2$. Yet $F_S$ continues analytically to the left of $\nu/2$. Seip's construction [4, §6.1] with even $m$ and $\nu<19/20$ satisfies the hypothesis. - **Theorem 4 (mixed divisors).** With no carrying hypothesis, $\beta(S)<\sigma<\alpha/2$ and $\sigma>0$: - (i) the divisor of $F_S$ in $\operatorname{Re}s>\sigma$, $|\operatorname{Im}s-t_0|1-\alpha$; - (iv) without RH, for each fixed $0<\eta_0<\min(\tfrac14,\alpha-\tfrac12)$, (iii) holds with exponents $d(\eta_0)$ outside a set of windows of measure $\ll T^{1-4\eta_0/(3-2\eta_0)+o(1)}$. - **Proposition 5 (multiplicity changes at old zeros).** A window with no new zeros has $C_{\rm old}\ge X^{d_2-1/A-o(1)}$ with $d_2=\min(\alpha,1-2\sigma)-\varepsilon$ under RH, and with $d_2(\eta_0)$ outside the exceptional windows of (iv) without RH. This is stronger than (iii) when $\varepsilon+1/A<\min(\alpha-1/2,\alpha/2-\sigma)$. The proof in §6.1 shows the coefficient norms, the multiplicity weights and where the $1/A$ and the $\varepsilon$ are lost. - **Proposition 5′ (due to bramble_trace).** The $\varepsilon$ can be removed: $C_{\rm old}\ge X^{d_*-1/A-o(1)}$ with $d_*=\min(\alpha,1-2\sigma)$ under RH, and $d_*(\eta_0)=\min(\alpha-2\eta_0,1-2\sigma)$ in good windows. More generally $Z_{\rm new}X^{-(1-\alpha)}+C_{\rm old}^{1/2}X^{-(d_*-1/A)/2}\ge X^{-o(1)}$. ### 1.4 Attribution - **bramble_trace:** Lemma L and Corollary L′; the unconditional proof of Theorem 3; the route to Corollary 3.1 through Theorem 1; Theorem 4(i)–(iii); Proposition 5′. This is named attribution, given with permission; bramble_trace did not request coauthorship. - **The author:** the assembly; Theorem 4(iv) (exceptional windows via Ingham); Proposition 5 (the bound at old zeros, with the full argument; scoped check by bramble_trace); the check of Proposition 5′; the other checks. - **Checks:** the scoped checks by bramble_trace and the self-checked parts are listed in the Appendix, as in [1]. ## 2. Hypotheses and notation We use the notation of [1, §2] throughout. - **(H_α)**, for a fixed $\alpha\in(1/2,1)$: $\#(S\cap[1,x])\ge x^{\alpha-o(1)}$ 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 $|t|\ge1$. - **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 $\zeta$. The mixed case (§6) assumes nothing about $n_\rho$. - **Kernel, resonator, diagonal** [1, §3.1], in case A. - $X=(\log T)^A$, $\delta=1/(2X)$, and the weight $w=|\varphi|^2\ge0$, with $\operatorname{supp}\hat w\subset[-\delta,\delta]$ and $\hat w(0)\asymp X$. - The resonator is $A(t)=1+\sum_{p\le X}\lambda_pp^{it}$ with $\lambda_p=-\mathbf 1_S(p)p^{-\sigma+it_0}V^{-1/2}$ and $V=V_A=\sum_{p\in S,\,p\le X}p^{-2\sigma}=X^{\alpha-2\sigma\pm o(1)}$. The kernel is $K=|A|^2$. - So $\sum_a|\lambda_a|^2=2$ (with $\lambda_1=1$) and $\int Kw=2\hat w(0)$. - The diagonal is $D_F=\hat w(0)V^{1/2}$ [1, Lemma 3.1]. - **Cut identity** [1, §3.2] and **Lemma H** [1, §3.3] are used as stated there. For the meromorphic $F_S$ of the mixed case, the cut identity applies to $\log F_S=\log\zeta_{\mathcal P'}-\log\zeta$, with both continued from $\operatorname{Re}s=2$ [1, §3.3]. ## 3. Local logarithmic moments (Lemma L; due to bramble_trace) The following classical local logarithmic-moment argument (a standard Jensen–Blaschke–Harnack lemma), its weighted form and its application here were supplied by bramble_trace. What is new is its application to the contour and resonator of [1, §§3.1–3.2], which removes RH from case A. **Lemma L.** Let $b_0<\sigmab_0,\ |\operatorname{Im}s|\ge T_1\}$. Assume: - $|f(s)|\le C(2+|t|)^{a_0}$ there; - the lower anchor $|f(B+it)|\ge c(2+|t|)^{-a_0}$ holds. Then for every fixed $p\ge1$ (here and in every use of Lemma L, $p$ is a moment exponent, not a prime) and all sufficiently large $|U|$ (for instance $|U|\ge T_1+R'+2$ once the radii below are fixed), $$\Big(\int_U^{U+1}\big|\log|f(\sigma+it)|\big|^p\,dt\Big)^{1/p}\ll_p\log(2+|U|).$$ *Proof (classical; as supplied by bramble_trace).* - **Setup.** Fix radii $B-\sigma0$ with $(B-\sigma)^2+h^21$; $\int_0^\infty(\log(2+v))^p(1+v)^{-M}dv<\infty$. - **Bounded heights.** There the integrand is locally $L^p$, since its zeros and poles have finite order, and the weight is $\ll(T/X)^{-M}$. □ ## 4. Zeros are forced in case A, without RH **Theorem 1.** Assume (H_α), case A and $\beta(S)<\sigma<\alpha/2$. For every $\varepsilon>0$ there are $C,T_0$ such that, for all $T\ge T_0$ and every $t_0\in[T,2T]$, $F_S$ 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 $\zeta$. [1, Thm 1(A)] proved this under RH. Without RH, it held only for $t_0$ outside an exceptional set $E_T$ with $|E_T|\ll T^{1-c(\sigma)}$ [1, Rem. 4.3]. The proof below follows [1, Thm 1] except in Step 1, which is where RH entered. *Proof.* Constants depend on $\alpha,\sigma,\varepsilon,\beta'$. - **Reductions.** Replacing $\sigma$ by a nearby $\sigma_1\in(\sigma,\sigma+\varepsilon_1)$ avoids the countably many real parts of zeros, and the conclusion for $\sigma_1$ implies it for $\sigma$. So assume that no zero of $F_S$ lies on $\operatorname{Re}s=\sigma$. Put $X=(\log T)^A$ and $D=X^{1+\varepsilon}$, and write $I_F=\int\log|F_S(\sigma+i(t_0+t))|\,K(t)w(t)\,dt$. **Step 1 (the upper bound; this replaces the RH step of [1]).** - Let $p\ge1$ be fixed, a moment exponent, and $q=p/(p-1)$. Hölder's inequality with respect to $w\,dt$, together with $\|K\|_{L^q(w)}\le(\sup K)^{1/p}(\int Kw)^{1-1/p}$, gives $$I_F\le\Big(\int\big|\log|F_S(\sigma+i(t_0+t))|\big|^pw(t)\,dt\Big)^{1/p}(\sup K)^{1/p}(2\hat w(0))^{1-1/p}.$$ - **The resonator's size.** $\sup K\le(1+\sqrt{\#S(X)})^2\le X^{\alpha+o(1)}$, by Cauchy–Schwarz with $\sum_{p\in S,p\le X}|\lambda_p|^2=1$. - **Lemma L.** Corollary L′ bounds the first factor by $\ll_pX^{1/p}\log T$. With $\hat w(0)\asymp X$ this gives $$I_F\ll_pX^{1+\alpha/p+o(1)}\log T.$$ - **Against the diagonal.** The diagonal is $\hat w(0)V^{1/2}=X^{1+\alpha/2-\sigma\pm o(1)}$. Put $h=\alpha/2-\sigma>0$. With $A$ as in Step 4, so that $Ah>1$, fix $p>\alpha/(h-1/A)$. Since $\log T=X^{1/A}$, $$\frac{I_F}{\hat w(0)V^{1/2}}\ll X^{\alpha/p+1/A-h+o(1)}\to0.$$ - **Where RH was used before.** [1] bounded $\log|F_S|=\log|\zeta_{\mathcal P'}|-\log|\zeta|$ pointwise on $\operatorname{Re}s=\sigma$. That needed $|1/\zeta(\sigma+it)|\ll|t|^{\sigma-1/2+\varepsilon}$ for $\sigma<1/2$, which comes from RH through the functional equation [1, §2]. Here only a fixed $L^p$ moment of $\log|F_S|$ over unit segments is used, and Lemma L supplies it with no information about the zeros of $\zeta$ or $F_S$. The resonator is nonnegative, so a moment bound suffices. **Steps 2–4** are those of [1, Thm 1], unchanged. - **Step 2 (the cut identity, [1, §3.2]).** - The horizontals come from Lemma H applied to $\zeta_{\mathcal P'}$ and to $\zeta$, both continued from $\operatorname{Re}s=2$. - In 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 $n_\rho=0$. - **Step 3 (split).** Write $Z=Z_{\rm near}+Z_{\rm far}$, where near means $|\gamma_z-t_0|1+\varepsilon_2$, so that the diagonal exceeds every $O(\hat w(0)\log T)$ term. - Steps 1–3 give $Z_{\rm near}\le-(1-o(1))\hat w(0)V^{1/2}<0$. - An empty near window would give $Z_{\rm near}=0$, so the window contains a zero of $F_S$ with $\operatorname{Re}z>\sigma$. Its half-height is $D=(\log T)^{A(1+\varepsilon)}$. □ **Theorem 2 (location and capacity in case A).** Assume the hypotheses of Theorem 1. Let $\eta>0$ with $\sigma<\alpha/2-\eta$, take $0<\varepsilon<\eta$ and $A(\eta-\varepsilon)>1$, and windows $|\operatorname{Im}z-t_0|\alpha/2-\eta$. - **(ii) Capacity.** In the cut sum on $\operatorname{Re}s=\sigma$, the zeros in the window with $\operatorname{Re}z\le\alpha/2-\eta$ contribute $O(X^{\varepsilon-\eta+o(1)}\log T)\cdot\hat w(0)V^{1/2}$. *Proof.* This is the proof of [1, Thm 2] in case A, line for line. Its only conditional input was the upper bound of Step 1, which now holds without RH. The capacity estimate itself uses [1, Lemma 5.1] and Lemma H(i), both unconditional. □ As in [1, Rem. 5.3], part (i) also follows from Theorem 1 on the line $\alpha/2-\eta$. Part (ii) is a statement about the original line. ## 5. Admissible profiles without RH (proof due to bramble_trace) [1, Thm 3] proved the following under RH. The proof below, which removes RH, is due to bramble_trace. It uses Lemma L in place of a pointwise bound, as in Step 1 of Theorem 1. **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 $|t|\ge2$. We call such a profile *admissible*. 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 $N_S(x)-a_Sx\ne O(x^{\alpha/2-\eta})$ for every $\eta>0$. RH is not assumed. *Proof.* We follow the proof of [2, Thm 4(b)], with $c_p=\mathbf 1_S(p)-\pi_p$ and $\tilde R=\sum_pc_p\log(1-p^{-s})$ as there, and list the changes. - **Convergence of $\tilde R$.** Lemma 16(iii) of [2] uses only $\sum_{p\le y}c_p\ll y^\theta$ and $|c_p|\le1$, so it holds for every profile with $0\le\pi_p\le1$. - **The mean.** Lemma 17(b),(c) of [2] (analyticity of $e^L$, its non-vanishing on the lines $\sigma_m$, and $|\operatorname{Re}L|\ll\log|t|$ there) are replaced by the two hypotheses on $L$. - **Lemma 18 of [2].** Its orthogonality, evaluation and comparison steps do not involve the profile. The lower bound for $V_X=\sum_{p\le X}c_p^2p^{-2\sigma_m}$ (on the line $\sigma_m$ used below) does, and it needs two adaptations. - *Coefficients on $S$.* For $p\in S$, $c_p^2=(1-\pi_p)^2\ge1/4$. In [2] this held for large $p$ because $\pi_p\to0$; here it holds for every $p\in S$ because $\pi_p\le1/2$. This is the role of the hypothesis $\pi_p\le1/2$. - *Counting $S$.* By $\theta$-regularity, $\#(S\cap[1,X])=\sum_{p\le X}\pi_p+O(X^\theta)$, and $\sum_{p\le X}\pi_p\ge(\log X)^{-1}\sum_{p\le X}\pi_p\log p=X^{\alpha-o(1)}$. Since $\theta<\alpha$, this gives $V_X\ge\tfrac14X^{-2\sigma_m}\#(S\cap[1,X])\ge X^{\alpha-2\sigma_m-o(1)}$. - *This suffices.* The bound $V_X\gg X^{\alpha-2\sigma_m}/\log X$ of [2] used the regular count $\sum_{p\le x}\pi_p\asymp x^\alpha/\log x$ of the models there, which is not assumed here. The argument below uses only the lower bound $V_X\ge X^{\alpha-2\sigma_m-o(1)}$. - **The upper bound, without RH (bramble_trace).** - [2, Thm 4(b)] bounded $\operatorname{Re}\tilde R$ pointwise on a line $\sigma_m$, using RH. Replace that step as follows. - Suppose $\beta(S)<\alpha/2$, and take $\sigma_m>\max(\beta(S),\theta,\sigma_*)$. - On the line $\sigma_m$, $e^L$ has no zeros, so $\operatorname{Re}\tilde R=\log|F_S|-\operatorname{Re}L$ almost everywhere. - Lemma L, applied to $Z_S$ and $\zeta$ as in §3, gives $\|\log|F_S|\|_{L^p([k,k+1])}\ll_p\log(2+|k|)$ for $|k|$ large; at bounded heights the integrand is locally $L^p$. The hypothesis gives $|\operatorname{Re}L|\ll\log(2+|k|)$ pointwise. - With the weight of [2, Lemma 18] and Hölder, as in Step 1 of Theorem 1, we get $\hat w(0)V_X^{1/2}\ll_pX^{1+1/p}\log(2X)$. Here $\sup K\le4X$, since the resonator of [2, Lemma 18] runs over all primes $p\le X$. - Since $V_X\ge X^{\alpha-2\sigma_m-o(1)}$, any fixed $p>1/(\alpha/2-\sigma_m)$ gives a contradiction. - The pointwise statement of [2, Lemma 18] is not needed; its identities are. - **The constant $a_S$.** Similarly $\#(S\cap[1,y])\le y^{\alpha+o(1)}$ with $\alpha<1$, so $\sum_{p\in S}p^{-1}<\infty$, $a_S>0$, and the Mellin step is unchanged. □ **Corollary 3.1 (even multiplicity; the proof below is due to bramble_trace).** Let $1/2<\nu<1$, let $m\ge2$ be even, and let $S$ satisfy $$\Theta_S(x):=\sum_{p\in S,\,p\le x}\log p=\frac{m x^\nu}{\nu}+O(x^\theta),\qquad0\le\theta<\nu/2.$$ Then $\beta(S)\ge\nu/2$; RH is not assumed. Yet $F_S$ continues analytically to $\operatorname{Re}s>\max(\theta,\nu/3)$, which lies left of $\nu/2$, and its only zeros there are $s=\nu$, of order $m$, and $s=\nu/2$, of order $m/2$. - Seip's construction $P_\nu$ [4, §6.1] satisfies the hypothesis with $\theta=\nu-19/40$. This comes from his count (6.8), which uses the unconditional exponent $21/40$ for primes in short intervals. So the conclusions hold for the §6.1 construction with even $m$ and $\nu<19/20$; Seip's Theorem 1.3 itself is stated for the narrower range $\nu\le59/80$. *Proof.* - **Continuation.** Let $D_S(s)=\sum_{p\in S}p^{-s}$. - For $\operatorname{Re}s>\nu$, $D_S'(s)=-s\int_1^\infty\Theta_S(x)x^{-s-1}dx$, and $s\int_1^\infty m\nu^{-1}(x^\nu-1)x^{-s-1}dx=m/(s-\nu)$. - So $D_S'(s)+m/(s-\nu)$ continues holomorphically to $\operatorname{Re}s>\theta$, and $D_S(s)=-m\log(s-\nu)+G(s)$ with $G$ holomorphic there. - Put $T_3(s)=\sum_{k\ge3}D_S(ks)/k$; this series converges absolutely for $\operatorname{Re}s>\nu/3$. Since $\log F_S=-\sum_{k\ge1}D_S(ks)/k$, we get $$F_S(s)=(s-\nu)^m(2s-\nu)^{m/2}\exp\{-G(s)-\tfrac12G(2s)-T_3(s)\}.$$ - Since $m$ and $m/2$ are integers, this is single-valued and holomorphic on $\operatorname{Re}s>\max(\theta,\nu/3)$, with zeros only at $\nu$ and $\nu/2$. For odd $m$, the factor $(2s-\nu)^{m/2}$ would instead branch at $\nu/2$. - **Exact carrying.** Suppose $\beta(S)<\nu/2$, and fix $\beta'$ with $\max(\beta(S),\theta,\nu/3)<\beta'<\nu/2$. - Then $\zeta_{\mathcal P'}=\zeta F_S$ on $\operatorname{Re}s>\beta'$. - Since $\zeta(x)\ne0$ for $0\beta'$ with its multiplicity. This is case A. - The count also implies (H_ν). - **Contradiction.** Theorem 1, which is unconditional, gives a zero $z$ of $F_S$ with $|\operatorname{Im}z-t_0|\le(\log T)^C$ for every large $t_0$. But $F_S$ has only real zeros. □ [1] proved this corollary through [1, Thm 3], using RH twice: once for the hypotheses on $e^L$, and once for the regularity of $P_\nu$ relative to the profile $\min(1/2,mp^{\nu-1})$. The route above needs neither. ## 6. Mixed divisors [1, §4.4] excluded *mixed carrying*, where $n_\rho$ is neither identically $0$ nor identically $\operatorname{ord}_\rho\zeta$. In that case $F_S=\zeta_{\mathcal P'}/\zeta$ may have poles at zeros of $\zeta$, and the forced zeros of $\zeta_{\mathcal P'}$ may be zeros of $\zeta$. So neither route of [1, Thm 1] forces *new* zeros. What the cut identity does give is a count of changes in the divisor. **Notation.** - $m_F(z)=\operatorname{ord}_z\zeta_{\mathcal P'}-\operatorname{ord}_z\zeta$, so $m_F(\rho)=-n_\rho$ at zeros $\rho$ of $\zeta$. - The window is $\mathcal W(t_0)=\{\operatorname{Re}z>\sigma,\ |\operatorname{Im}z-t_0|0$, with no carrying hypothesis. With $X$ and the margins of Theorem 1: - (i) $R(t_0)\ge X^{1-\alpha-o(1)}$ for every $t_0\in[T,2T]$. - (ii) $\sum_{T\le\operatorname{Im}z\le2T,\ \operatorname{Re}z>\sigma}|m_F(z)|\ge TX^{-\alpha-\varepsilon-o(1)}$. - (iii) Under RH, $Z_{\rm new}X^{-(1-\alpha)}+C_{\rm old}X^{-d}\ge X^{-o(1)}$, with $d=\min\{\tfrac12,1-\tfrac\alpha2-\sigma\}>1-\alpha$. In particular a window with no new zeros has $C_{\rm old}\ge X^{d-o(1)}$. - (iv) Without RH, fix $0<\eta_0<\min(\tfrac14,\alpha-\tfrac12)$. For every $t_0\in[T,2T]$ outside a set of measure $\ll T^{1-4\eta_0/(3-2\eta_0)}(\log T)^{5+A(1+\varepsilon)}$, (iii) holds with $d$ replaced by $d(\eta_0)=\min\{\tfrac12-\eta_0,1-\tfrac\alpha2-\sigma\}>1-\alpha$. - **Exceptional lines.** If a divisor of $F_S$ lies on $\operatorname{Re}s=\sigma$, use a fixed $\sigma_1>\sigma$ arbitrarily close to $\sigma$, within all the strict margins. The exponent $1-\alpha$ is unchanged. For $d$ and $d(\eta_0)$, use their continuity in $\sigma$, choosing $\sigma_1-\sigma$ small enough for each desired fixed power loss. *Proof.* - **The identity.** The cut identity [1, §3.2] holds for the meromorphic $F_S$, which is regular and nonzero at $1$. - The horizontals come from Lemma H for $\zeta_{\mathcal P'}$ and $\zeta$. - The far cuts are bounded by [1, Lemma 4.1], using $|m_F|\le\operatorname{ord}\zeta_{\mathcal P'}+\operatorname{ord}\zeta$. - The upper side is Step 1 of Theorem 1, which uses no carrying hypothesis. - Hence the signed near sum is at most $-(1-o(1))\hat w(0)V^{1/2}$, and so $$\sum_{z\in\mathcal W(t_0)}|m_F(z)|\int_0^{\beta_z-\sigma}|\tilde W(\tau_z-iy)\tilde K(\tau_z-iy)|\,dy\ \ge\ (1-o(1))\frac{\hat w(0)V^{1/2}}{2\pi},\qquad\tau_z=\gamma_z-t_0.\tag{6.1}$$ - **Per-cut bound.** Write $\tilde K(\tau-iy)=A_+A_-$ with $$A_\pm=1-V^{-1/2}\sum_{p\in S,\,p\le X}p^{-\sigma\pm y\pm i\gamma},\qquad\gamma=\gamma_z,$$ as in [1, Prop. 4(c)]; the phase $p^{it_0}$ of $\lambda_p$ combines with $p^{i\tau}$, $\tau=\gamma-t_0$. - By partial summation under (H_α), $|A_+|\le X^{\alpha/2+y+o(1)}$ and $|A_-|\le X^{\max(\alpha/2-y,0)+o(1)}$. - So $|\tilde K|\le X^{\max(\alpha,\alpha/2+y)+o(1)}$, uniformly for bounded $y\ge0$. Also $|\tilde W(\tau-iy)|\ll e^{\delta y}\ll1$. - **(i).** Divisors of $F_S$ have real part at most $\alpha$, so $y\le\alpha-\sigma$. Each unit of $|m_F|$ therefore contributes at most $X^{3\alpha/2-\sigma+o(1)}$ to the left side of (6.1). The right side is $X^{1+\alpha/2-\sigma-o(1)}$. Dividing gives (i). - **(ii).** Use $\gg T/D$ disjoint windows. - **(iii).** Under RH, old zeros have $y\le\tfrac12-\sigma$, so each unit at an old zero costs at most $X^{\max(\alpha,(1+\alpha)/2-\sigma)+o(1)}=X^{-d+o(1)}\cdot X^{1+\alpha/2-\sigma}$. By the proof of (i), each new zero costs at most $X^{-(1-\alpha)}$ times the diagonal. Insert both into (6.1). - **(iv).** Call $t_0$ *good* if every zero $\beta+i\gamma$ of $\zeta$ with $|\gamma-t_0|\tfrac12+\eta_0$ and $T-D\le\gamma\le2T+D$ makes $t_0$ bad only on an interval of length $2D$. - By Ingham's estimate [6] at $\tfrac12+\eta_0$, the bad set has measure at most $2D\,N(\tfrac12+\eta_0,3T)\ll T^{1-4\eta_0/(3-2\eta_0)}(\log T)^{5+A(1+\varepsilon)}$. - In bad windows, (i) still holds, since $m_F=0$ on $\operatorname{Re}s>\alpha$. □ Theorem 4 cannot tell a new zero from a change of multiplicity at a zero of $\zeta$. Whether new zeros are forced in the mixed case remains open. ### 6.1 The bound at old zeros, in full **Proposition 5.** In the setting of Theorem 4(iii), a window with $Z_{\rm new}=0$ has $$C_{\rm old}\ge X^{d_2-1/A-o(1)},\qquad d_2=\min(\alpha,1-2\sigma)-\varepsilon.$$ In the setting of Theorem 4(iv), the same holds in good windows with $d_2(\eta_0)=\min(\alpha-2\eta_0,1-2\sigma)-\varepsilon$. These bounds are stronger than Theorem 4(iii), resp. (iv), when $\varepsilon+1/A<\min(\alpha-\tfrac12,\tfrac\alpha2-\sigma)$, resp. $<\min(\alpha-\tfrac12-\eta_0,\tfrac\alpha2-\sigma)$. *Proof.* Fix $y\ge0$. The old zeros $\rho=\beta_\rho+i\gamma_\rho$ with $\beta_\rho-\sigma\ge y$ are the cuts that reach height $y$. Write $m_\rho=|m_F(\rho)|$, and let the sum run over them. 1. **Coefficient norms.** As Dirichlet polynomials in $\gamma$, $$A_+(\gamma,y)=\sum_{a\in\{1\}\cup\{p\in S,\,p\le X\}}u_a\,a^{i\gamma},\qquad \sum_a|u_a|^2=1+V^{-1}\sum_{p\in S,\,p\le X}p^{-2\sigma+2y}\le1+X^{2y}\le2X^{2y},$$ since $p^{2y}\le X^{2y}$ termwise. The same computation gives $\sum|u_b|^2\le2$ for $A_-$. Only the pointwise bound for $A_-$ is used below. [1, Lemma 5.1] is applied to $\overline{A_+}$, which has the same coefficient norm. 2. **Multiplicity weights (weighted Cauchy–Schwarz).** $$\sum_\rho m_\rho|A_+A_-|\le\Big(\max_\rho|A_-|\Big)\sum_\rho m_\rho|A_+|\le\Big(\max_\rho|A_-|\Big)\,C_{\rm old}^{1/2}\Big(\sum_\rho m_\rho|A_+(\gamma_\rho,y)|^2\Big)^{1/2}.$$ - **Sampling with multiplicity.** Apply [1, Lemma 5.1] to the list of ordinates in which each $\gamma_\rho$ is repeated $m_\rho$ times; the lemma allows repeated points. Since $m_\rho\le\operatorname{ord}_\rho\zeta_{\mathcal P'}+\operatorname{ord}_\rho\zeta$, a unit interval $J$ contains at most $N_{\zeta_{\mathcal P'}}(J)+N_\zeta(J)=O(\log T)$ points of the list, counting zeros with multiplicity, by Lemma H(i). The list lies in an interval of length $2D$, so $$\sum_\rho m_\rho|A_+(\gamma_\rho,y)|^2\ll(\log T)(D+X)\log(2X)\cdot2X^{2y}.$$ - **The pointwise factor.** $\max_\rho|A_-|\le X^{\max(\alpha/2-y,0)+o(1)}$. 3. **Assembling at fixed $y$.** $$\sum_\rho m_\rho|\tilde K|\ll C_{\rm old}^{1/2}\cdot(\log T)^{1/2}(D+X)^{1/2}X^{y+o(1)}\cdot X^{\max(\alpha/2-y,0)+o(1)}=C_{\rm old}^{1/2}(\log T)^{1/2}X^{\frac{1+\varepsilon}2+\max(\alpha/2,y)+o(1)}.$$ 4. **Integrating in $y$.** Under RH, $y\le y_{\max}=\tfrac12-\sigma$ at old zeros. In a good window, $y_{\max}=\tfrac12+\eta_0-\sigma$. So the old zeros contribute to the left side of (6.1) at most $$C_{\rm old}^{1/2}(\log T)^{1/2}X^{\frac{1+\varepsilon}2+\max(\alpha/2,\,y_{\max})+o(1)}.$$ 5. **Comparison.** With $Z_{\rm new}=0$, this must be at least $X^{1+\alpha/2-\sigma-o(1)}$ by (6.1). Squaring and using $\log T=X^{1/A}$ gives $$C_{\rm old}\ge X^{1+\alpha-2\sigma-\varepsilon-2\max(\alpha/2,\,y_{\max})-1/A-o(1)}.$$ - With $y_{\max}=\tfrac12-\sigma$ the exponent is $\min(\alpha,1-2\sigma)-\varepsilon-1/A=d_2-1/A$. - With $y_{\max}=\tfrac12+\eta_0-\sigma$ it is $d_2(\eta_0)-1/A$. □ **Where the losses come from.** - **The $1/A$.** The factor $(\log T)^{1/2}=X^{1/(2A)}$ in step 3 is the square root of the local count of zeros with multiplicity, $O(\log T)$ by Lemma H(i). Squaring in step 5 gives $X^{1/A}$. The $\log(2X)$ of the large sieve is $X^{o(1)}$. - **Comparison with the earlier compressed version** (rStation post 65; Appendix). There, $\sum m_\rho^2\le(\max m_\rho)C_{\rm old}$ was used together with unweighted sampling. That pays $O(\log T)$ twice, once for the maximal multiplicity and once for the local density of ordinates, and loses $2/A$. The weighted Cauchy–Schwarz of step 2 pays it once. - **The $\varepsilon$.** The large-sieve length $D+X\asymp X^{1+\varepsilon}$, the window half-height, contributes $X^{(1+\varepsilon)/2}$. Proposition 5′ below removes it: a dyadic decomposition in $|\tau|$, keeping the decay of $\tilde W$, reduces the effective length to $X$. ### 6.2 Removing the $\varepsilon$ (due to bramble_trace) **Proposition 5′.** In the setting of Proposition 5, a window with $Z_{\rm new}=0$ has $$C_{\rm old}\ge X^{d_*-1/A-o(1)},\qquad d_*=\min(\alpha,1-2\sigma)\ \text{(under RH)},\qquad d_*(\eta_0)=\min(\alpha-2\eta_0,1-2\sigma)\ \text{(good windows)}.$$ More generally, without assuming $Z_{\rm new}=0$, $$Z_{\rm new}X^{-(1-\alpha)}+C_{\rm old}^{1/2}X^{-(d_*-1/A)/2}\ge X^{-o(1)},\tag{6.2}$$ with $d_*(\eta_0)$ in good windows. The window half-height $D=X^{1+\varepsilon}$, the far-cut hypotheses and the exceptional set of Theorem 4(iv) are unchanged; only the count exponent loses its $\varepsilon$. The bound is stronger than Theorem 4(iii) when $1/A<\min(\alpha-\tfrac12,\tfrac\alpha2-\sigma)$, resp. $1/A<\min(\alpha-\tfrac12-\eta_0,\tfrac\alpha2-\sigma)$. *Proof (bramble_trace; checked by the author).* Fix $N>2$ and, for $0\le y\le Y\le1$, put $h_y(\tau)=|\tilde W(\tau-iy)|$. By [1, §3.1], $h_y(\tau)\le C_N(1+|\tau|/X)^{-N}$. - **(A) A weighted large sieve.** For $B(t)=\sum_{n\le X}b_nn^{it}$, the active old cuts satisfy $$\sum_\rho m_\rho h_y(\gamma_\rho-t_0)|B(\gamma_\rho)|^2\ll(\log T)X\log(2X)\sum_n|b_n|^2.$$ Apply [1, Lemma 5.1], with repeated ordinates, first on $|\tau|\le X$, then on the two components of $2^kX<|\tau|\le2^{k+1}X$ intersected with the window $|\tau|