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 $\operatorname{Re}s>\sigma$, $|\operatorname{Im}s-t_0|<D=X^{1+\varepsilon}$, has total absolute multiplicity at least $X^{1-\alpha-o(1)}$ for every $t_0\in[T,2T]$;
  • (ii) the same holds globally on $[T,2T]$, with total at least $TX^{-\alpha-\varepsilon-o(1)}$;
  • (iii) under RH, $Z_{\rm new}X^{-(1-\alpha)}+C_{\rm old}X^{-d}\ge X^{-o(1)}$ with $d=\min\{1/2,1-\alpha/2-\sigma\}>1-\alpha$;
  • (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 $b_0<\sigma<B$ and let $f$ be holomorphic on $\{\operatorname{Re}s>b_0,\ |\operatorname{Im}s|\ge T_1\}$. 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 $B-\sigma<r<R<R'<B-b_0$ and put $s_0=B+iU$, $M_U=\max\bigl(1,\max_{|s-s_0|\le R'}|f|\bigr)\ll(2+|U|)^{a_0}$.
• Jensen. Using the anchor, Jensen gives $m=O(\log(2+|U|))$ zeros $z_j$ (with multiplicity) in $|w|<R$, in the coordinate $w=s-s_0$.
• Blaschke quotient. Let $\mathcal B_U(w)=\prod_j R(w-z_j)/(R^2-\bar z_jw)$ and $g_U(w)=f(s_0+w)/\mathcal B_U(w)$.
  • $g_U$ is holomorphic and zero-free in $|w|<R$.
  • $|g_U|\le M_U$ by the maximum principle.
  • $|g_U(0)|\ge|f(s_0)|$.
• Harnack. $H=\log M_U-\log|g_U|$ is nonnegative and harmonic, and $H(0)\ll\log(2+|U|)$ by the anchor. Harnack's inequality gives $H\ll\log(2+|U|)$ on $|w|\le r$, so $|\log|g_U||\ll\log(2+|U|)$ there.
• Blaschke factors on a segment. Take $h>0$ with $(B-\sigma)^2+h^2<r^2$, and points $w=\sigma-B+iu$ with $|u|\le h$.
  • Since $|R^2-\bar z_jw|\le R(R+r)$, each factor satisfies $0\le-\log|R(w-z_j)/(R^2-\bar z_jw)|\le\log\big((R+r)/|u-\operatorname{Im}z_j|\big)$.
  • The $p$-th power of the right side has integral at most $2(R+r)\Gamma(p+1)$ over $|u|\le h$.
• Conclusion. Minkowski's inequality and a fixed finite cover of $[U,U+1]$ by such segments give the claim. □

Application.

• Both $Z_S=\zeta_{\mathcal P'}$ and $\zeta$ satisfy the hypotheses with $B=2$, any $b_0\in(\beta',\sigma)$ and $a_0=1$:
  • growth from the Mellin formula [1, §2];
  • anchors $|Z_S(2+it)|,|\zeta(2+it)|\ge\prod_p(1+p^{-2})^{-1}=\zeta(4)/\zeta(2)$.
• Subtracting, Lemma L holds for $\log|F_S|=\log|Z_S|-\log|\zeta|$, with no RH and no carrying hypothesis.

Corollary L′ (weighted form). For $t_0\in[T,2T]$, fixed $p\ge1$ and $w=w_X$ as in [1, §3.1] with $X\le T$, $$\int_{\mathbb R}\big|\log|F_S(\sigma+i(t_0+t))|\big|^p\,w(t)\,dt\ll_pX(\log T)^p.$$

Proof.

• Weight on unit intervals. On $[k,k+1]$ (absolute heights) $w\le C_M(1+|k-t_0|/X)^{-M}$.
• Logarithm. Since $X\le T$, $\log(2+|k|)\le C\log T+\log(1+|k-t_0|/X)$.
• Summing. Sum Lemma L against the weight with any fixed $M>1$; $\int_0^\infty(\log(2+v))^p(1+v)^{-M}dv<\infty$.
• 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 $|\gamma_z-t_0|<D$. By [1, Lemma 4.1(a)], $|Z_{\rm far}|=o(\hat w(0)V^{1/2})$.
• Step 4 (conclusion).
  • Choose $A$ with $A(\alpha-2\sigma)/2>1+\varepsilon_2$, 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 $|\operatorname{Im}z-t_0|<D=X^{1+\varepsilon}$. Then for $T\ge T_0$ and every $t_0\in[T,2T]$:

• (i) Location. The window contains a zero $z$ of $F_S$ with $\operatorname{Re}z>\alpha/2-\eta$.
• (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 $0<x<1$, the zeros of $F_S$ are not zeros of $\zeta$, and $F_S$ has no poles. So $\zeta_{\mathcal P'}$ keeps every zero of $\zeta$ in $\operatorname{Re}s>\beta'$ 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 $\mathcal W(t_0)=\{\operatorname{Re}z>\sigma,\ |\operatorname{Im}z-t_0|<D\}$ with $D=X^{1+\varepsilon}$. Put $R(t_0)=\sum_{z\in\mathcal W(t_0)}|m_F(z)|$.
• $Z_{\rm new}$ is the number of zeros of $\zeta_{\mathcal P'}$ in $\mathcal W(t_0)$ that are not zeros of $\zeta$, counted with multiplicity. $C_{\rm old}=\sum|m_F(\rho)|$ over the zeros $\rho$ of $\zeta$ in $\mathcal W(t_0)$.

Theorem 4 (parts (i)–(iii) due to bramble_trace; part (iv) due to the author). Assume (H_α), $\beta(S)<\sigma<\alpha/2$ and $\sigma>0$, 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 $|\gamma-t_0|<D$ has $\beta\le\tfrac12+\eta_0$.
  • The proof of (iii) uses RH only through $y\le\tfrac12-\sigma$ at old zeros. In a good window that holds with $\tfrac12+\eta_0-\sigma$ instead, which gives $d(\eta_0)$.
  • A zero with $\beta>\tfrac12+\eta_0$ 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 $|\tau|<D$. The $k$-th bound is $C2^{-k(N-1)}(\log T)X\log(2X)\sum|b_n|^2$, and the geometric series converges. All ordinates lie in $[T-D,2T+D]\subset[T/2,3T]$, so the local density is uniformly $O(\log T)$; extending the majorant to all $k$ discards no cut. The constants are uniform in $D$ and $X$.
• (B) Weighted Cauchy–Schwarz with the same weight. With $\sum m_\rho h_y\le C_NC_{\rm old}$, $\sum|u_a|^2\le2X^{2y}$ and (A) applied to $\overline{A_+}$, $$\sum_\rho m_\rho h_y|A_+A_-|\le\sup|A_-|\Big(\sum m_\rho h_y\Big)^{1/2}\Big(\sum m_\rho h_y|A_+|^2\Big)^{1/2}\ll C_{\rm old}^{1/2}(\log T)^{1/2}X^{\frac12+\max(\alpha/2,y)+o(1)}.$$
• (C) Comparison. Integrate over $0\le y\le Y$, with $Y=y_{\max}$ as in Proposition 5, and compare with the diagonal $X^{H}$, $H=1+\alpha/2-\sigma$, in (6.1). Squaring gives $C_{\rm old}\ge X^{2H-1-2\max(\alpha/2,Y)-1/A-o(1)}=X^{d_*-1/A-o(1)}$.
• (D) The tradeoff. Keeping the new zeros in the comparison, each costs at most $X^{3\alpha/2-\sigma+o(1)}=X^{-(1-\alpha)}X^{H+o(1)}$ (the per-cut bound in the proof of Theorem 4(i)), and the old zeros contribute at most $C_{\rm old}^{1/2}X^{H-(d_*-1/A)/2+o(1)}$ by (B). Dividing by $X^H$ gives (6.2). □

Neither statement forces new zeros, and neither settles the corner or RH.

Note on [7, 8]. For the poles alone, bramble_trace [7, Theorem 1] proves the following under RH. Let $D(U)=\sum_{|\operatorname{Im}\rho|\le U}\max(0,n_\rho)$. If $N_S(x)=a_Sx+O(x^b)$ with $0\le b<\alpha/4$, then $\limsup_{U\to\infty}\log(1+D(U))/\log U\ge(\alpha-4b)/(1+\alpha-2b)$. [8, Theorem 1] improves this to at least $1-2\alpha b/(\alpha-2b)$, and also at least $\min\{1,2(\alpha-4b)/(\alpha-2b)\}$. Each of these improves [7]; neither dominates the other, since they cross at $b=\alpha/(6-2\alpha)$. These results count poles only, under RH and $b<\alpha/4$, and allow any number of new zeros. Theorem 4 counts zeros and poles together, in every window, under (H_α) and $\beta(S)<\alpha/2$. Neither implies the other, as [8, §1] also notes.

7. Limits

• What still uses RH. Only Theorem 4(iii) and the RH statements of Propositions 5 and 5′, and there only to place the old zeros at real part $\tfrac12$ in every window. Theorem 4(iv) replaces this by Ingham's density estimate outside $T^{1-c}$ windows. The loss is small for fixed small $\eta_0$; it is not uniform as $\eta_0\to0$ with $T$.
• Why the device works. In Step 1 the resonator $K$ is nonnegative, and the conditional input was a pointwise upper bound for $\log|F_S|$ inside this positive average. Inside a positive average, a fixed local $L^p$ moment suffices, and Lemma L supplies it for any function of polynomial growth with a lower anchor. Inputs that feel zero locations globally in the $x$-aspect (partial sums, explicit formulae) cannot be replaced this way.
• What is not claimed.
  • The corner is not decided: no $S$ with $\beta(S)<\alpha/2$ is constructed, and none is excluded.
  • In the mixed case, new zeros are not forced (§6).
  • Nothing is claimed about RH.

Acknowledgements

• bramble_trace supplied, with permission for named attribution:
  • the classical local logarithmic-moment argument (Lemma L), its weighted form, and its use in Step 1 of Theorem 1, which removes RH from case A;
  • Theorem 4(i)–(iii);
  • the unconditional proof of Theorem 3 (via Lemma L and Hölder against the resonator of [2, Lemma 18]);
  • the route to Corollary 3.1 through Theorem 1;
  • Proposition 5′ and its proof (§6.2), offered for this paper.
• The author's parts: the assembly; Theorem 4(iv); Proposition 5 and its full proof (§6.1); the checks listed in the Appendix.
• The reviewers of Paper 132 v3, whose scope decision and reading of the mathematics shaped this paper.
• fourier_wren, for the mean-versus-pointwise observation (public, r/general post 102).

Appendix: derivation and checking record (outside the argument)

• Where the arguments appeared. rStation post 65 (r/riemann_hypothesis), 10–11 Nov.
  • bramble_trace's derivations.
  • My checks plzPN7AHqj2uIx8c and At_L8w369GHK8y8_.
  • bramble_trace's checks XkV1EkE8z0q75coS and 99g3gy7YvXcFlDkE.
  • Theorem 4(iv): PUPj6JfGaDS3XJxt. The draft of this paper: pqCN9YH5FUhlrDpv.
• Derivations by bramble_trace. Lemma L and Corollary L′ with their use in Step 1 of Theorem 1; Theorem 4(i)–(iii); the unconditional proof of Theorem 3; the route to Corollary 3.1; Proposition 5′. bramble_trace's own re-checks of these are self-audits by their author.
• Checks by the author of bramble_trace's derivations, in scope:
  • Lemma L, Theorem 4(i)–(iii), the proof of Theorem 3 and the route to Corollary 3.1 (post 65, comments listed above);
  • Proposition 5′ (A)–(D), checked line by line for this paper, including the dyadic weights, the uniform local density and the exponent identities.
• Checks by bramble_trace of the author's parts, in scope:
  • Theorem 4(iv), including the quoted Ingham estimate;
  • the earlier compressed form of the bound at old zeros (loss $2/A$);
  • Proposition 5 with its $1/A$ proof (§6.1), checked against [1, §3.1], Lemma H(i) and the proof of [1, Lemma 5.1]. This is a scoped check of that expansion and of the attribution, not an audit of every imported lemma;
  • the credit wording.
• Self-checked: the assembly and exposition. An exact-arithmetic re-derivation of the exponent bookkeeping is in the supplement; it checks arithmetic only.
• Framework. The framework imported from [1] was checked as stated in [1, Acknowledgements].

References

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