Complex-only off-line zeros from deleting and doubling primes

Draft v0 (started 10:45 UTC, 7 Nov). Not for posting. Forum announcement: post 69.

1. Problem and results

For a set $S$ of primes, the deletion $f_S=\zeta\prod_{p\in S}(1-p^{-s})$ keeps an Euler product, a non-negative von Mangoldt function and integers inside $\mathbb N$. When $S$ has convergence exponent $\alpha\in(1/2,1)$ it violates the Riemann hypothesis. Paper [3] painted zeros at any prescribed height on the line $\operatorname{Re}s=\alpha$. Every painted deletion, however, keeps a dominating real zero at $\alpha$ [3, Cor. 1(b)], and every deletion with $\alpha>1/2$ has a real zero or singularity at $\alpha$ (Landau). So deletion controls cannot test global or low-height statements. Examples are "$\zeta$ has no zeros in $(1/2,1)$", RH verified to a finite height, and functionals dominated by the real point, such as Nyman–Beurling distances [3, §1].

This paper removes the real zero by allowing primes to be doubled as well as deleted. It also shows that the doubling is unavoidable: among Euler products over the rational primes with multiplicities, multiplicity one forces violations of RH onto the real axis first.

The construction.

• Fix $1/2<\alpha<1$ and put $\kappa=1-\alpha$.
• Let $m:\mathbb R\to\mathbb Z$ be finitely supported and symmetric, with $m(0)=0$, and put $g(u)=\sum_\gamma m(\gamma)e^{i\gamma u}$ and $M=\sum_\gamma|m(\gamma)|$. No positivity of $g$ is required.
• Fix $p_0$ with $Mp_0^{-\kappa}\leq1/8$ and $p_0\geq17$. For $p\geq p_0$ let $\varepsilon_p\in\{-1,0,+1\}$ be independent, with

$$ \mathbb P(\varepsilon_p=\pm1)=q_p^{\pm}:=\tfrac12p^{-\kappa}\bigl(M\mp g(\log p)\bigr),\qquad\text{so}\qquad\mathbb E\varepsilon_p=-p^{-\kappa}g(\log p). $$

• Put $\varepsilon_p=0$ for $p<p_0$, and define

$$ \zeta_\Pi(s)=\zeta(s)\prod_p\bigl(1-p^{-s}\bigr)^{-\varepsilon_p}=\prod_p\bigl(1-p^{-s}\bigr)^{-(1+\varepsilon_p)}. $$

• On $p_0$. Any $p_0$ with $Mp_0^{-\kappa}\leq1$ gives valid probabilities. The stronger condition above only simplifies uniform constants. For each prime, $\mathbb E(1-u)^{1-\varepsilon_p}=q_p^++(1-q_p^+-q_p^-)w+q_p^-w^2$, with $w=1-u$, is a quadratic in $w$ with non-negative coefficients. Its roots have $\operatorname{Re}w\leq0$, so it has no zeros for $|u|<1$. Hence finitely many primes below the stronger threshold contribute only bounded local factors to every estimate below. The numerics in §6 use $p_0=7$.
• This is the zeta function of a discrete Beurling system:
  • the generalized primes are the rational primes, each with multiplicity $1+\varepsilon_p\in\{0,1,2\}$, so $\Lambda_\Pi\geq0$;
  • the generalized integers are $\mathbb N$, with multiplicity $a(n)=\prod_{p^k\Vert n}\binom{\varepsilon_p+k}{k}\in\mathbb Z_{\geq0}$.

Theorem 1 (zeros; RH for $\zeta$). Almost surely, $\zeta_\Pi$ continues meromorphically to $\operatorname{Re}s>\alpha/2$. In that half-plane its zeros and poles are exactly:

• a zero of order $m(\gamma)$ at $\alpha+i\gamma$ (a pole of order $|m(\gamma)|$ if $m(\gamma)<0$), for each $\gamma$;
• a simple pole at $s=1$ with nonzero residue;
• the zeros of $\zeta$ on the critical line.

In particular, $m=\delta_\gamma+\delta_{-\gamma}$ gives simple zeros at $\alpha\pm i\gamma$ and no other zeros in $\operatorname{Re}s>1/2$; there is no real zero.

Theorem 2 (integers; RH). Let $1/2<\alpha\leq2/3$ and $m\geq0$. Almost surely $N(x):=\sum_{n\leq x}a(n)=ax+O(x^{\alpha/2+\varepsilon})$ with $a>0$. If moreover $m\not\equiv0$ (equivalently $M>0$), the exponent $\alpha/2$ is attained. (For $m\equiv0$ nothing is modified and $N(x)=\lfloor x\rfloor$.)

Theorem 3 (multiplicity one forces the real axis). Let $Z(s)=\prod_p(1-p^{-s})^{-n_p}$ with multiplicities $n_p\in\{0,1\}$. Let $D=\{p:n_p=0\}$ have convergence exponent $\sigma_D$.

• If $1/2<\sigma_D<1$, then $Z$ has a zero or a singularity at the real point $\sigma_D$.
• If $\sigma_D=1$, then $Z-c/(s-1)$ is analytic at $1$ for no constant $c\neq0$.

In every case $Z=\zeta e^{-L}$ is analytic and zero-free wherever $\zeta$ is, on $\operatorname{Re}s>\sigma_D$. So, under RH for $\zeta$, multiplicity one together with an exact nonzero residue means the following. First, $\sigma_D<1$, and every zero of $Z$ in $\operatorname{Re}s>1/2$ lies in $\operatorname{Re}s\leq\sigma_D$. Second, if there is such a zero, then $\sigma_D>1/2$ and the real point $\sigma_D$ carries a zero or a singularity. Theorem 1 shows that multiplicity two removes this.

Remark (what Theorem 3 uses). Multiplicity one is exactly $0\leq\Lambda_Z\leq\Lambda$, and the proof uses only $\Lambda-\Lambda_Z\geq0$, i.e. the positivity of the coefficients of $L$. The same proof with $-L$ in place of $L$ covers doubling alone ($n_p\in\{1,2\}$, $D=\{p:n_p=2\}$). For $1/2<\sigma_D<1$, $Z$ is then singular at $\sigma_D$; a zero is ruled out because $Z/\zeta=e^{L}\geq1$ on the real axis. For $\sigma_D=1$ the pole is not exact. So both signs are needed, as in the trichotomy of [N, §6].

Proposition 4 (primes; RH). Almost surely, $\psi_\Pi(x)=x-\sum_\gamma m(\gamma)x^{\alpha+i\gamma}/(\alpha+i\gamma)+O(x^{1/2}\log^2x)$. In particular there is no term $x^\alpha/\alpha$.

Proposition 5 (the integer barrier is sign-blind). Let $\varepsilon_p\in\{-1,0,1\}$ be any fixed choice of signs, not necessarily random. Suppose the modified set $S'=\{p:\varepsilon_p\neq0\}$ has convergence exponent $\alpha\in(0,1)$. Then for every $\varepsilon>0$, $N(x)-ax\neq O\bigl(x^{\alpha(1-\alpha)/(2(2-\alpha))-\varepsilon}\bigr)$, unconditionally. This is the barrier proved in [2, Thm A] for deletions. So doubling cannot be used to beat it, whereas for the zeros the signs decide everything (Theorem 3).

What is not claimed. Nothing is claimed about $\zeta$. The systems lack Riemann's functional equation and multiplicity one. By Theorem 3, giving up the second is necessary for complex-only violations over $\mathbb P$ with an exact simple pole. By the note [N], the first together with a simple pole forces $\zeta$.

2. Related work

• Deletion controls.
  • [1] constructs random sifting of $\mathbb P$ with integer error $x^{\alpha/2+\varepsilon}$.
  • [2] proves a universal barrier and a natural boundary at $\operatorname{Re}s=\alpha/2$.
  • [3] paints zeros on the abscissa line by deletion. Its Proposition 1 shows that deletion can realise exactly the patterns with $g\geq0$ and $m(0)\geq1$: the real zero dominates.
  • The present construction is the signed version. The positivity constraint disappears because doubled primes carry the negative part of the pattern.
• Prescribed zeros in general Beurling systems.
  • Broucke–Debruyne–Révész [B, Thm 3.2] prescribe any finite symmetric multiset of zeros and poles, with integer error $x^{1/2+o(1)}$ and a pole of high order at some $\delta<1/2$.
  • Their generalized primes are not rational primes. Relative to $\mathbb P$ their modification is signed, as the trichotomy of [N, §6] requires for complex-only zeros with an exact simple pole.
  • Here the primes are those of $\mathbb P$, each with multiplicity $0$, $1$ or $2$. The integers lie in $\mathbb N$, and under RH the integer error is $x^{\alpha/2+\varepsilon}$ for $\alpha\leq2/3$.
• Large oscillation. Diamond, Montgomery and Vorhauer [DMV] and Zhang [Z] built Beurling systems with well-behaved integers and zeros near $\operatorname{Re}s=1$. Révész [R] constructs, for a given zero $\rho_0$, a Beurling system whose prime error is at most $(\pi/2+\varepsilon)x^{\operatorname{Re}\rho_0}/|\rho_0|$. He uses a low-norm sine polynomial and the random approximation of Broucke–Vindas. Its primes are general, not rational. Beurling's example [Be] modulates the prime density by $1-\cos\log t$. Signed painting transplants the cosine modulation into multiplicities of rational primes.
• Zeros of products over a set of primes. Hilberdink and Saias [HS] study the zeros of $\sum_n\lambda_P(n)n^{-s}=\zeta_P(2s)/\zeta_P(s)=\prod_{p\in P}(1+p^{-s})^{-1}$ for a set $P$ of primes. Since $\zeta_P(2s)$ is analytic and zero-free on $\operatorname{Re}s>\sigma_P/2$, there these are the zeros of $\prod_{p\in P}(1-p^{-s})$, the deleted part in [3]. Their elementary results place the zeros in $0<\operatorname{Re}s\le\sigma_P$ and show that $1$ is a zero if and only if $\sum_{p\in P}1/p=\infty$. Their main theorem, a set $P$ whose zero set is a single prescribed simple real zero, was withdrawn by the authors in 2024 because of an error in its proof [HS, v2]. Deletion alone forces the real zero to dominate [3, Prop. 1]; the doubled primes here remove that constraint.
• Multiplicities on powers of rational primes. The zero-density counterexamples of Broucke [ZD] use generalized primes $p^{m/(1-\theta)}$ with integer multiplicities. Those are powers of rational primes, rescaled. Here the primes themselves are kept, and only their multiplicities change.
• The Landau trichotomy. Theorem 3 specialises the trichotomy of [N, §6] to multiplicities in $\{0,1\}$. That trichotomy says a one-signed modification of $\mathbb P$ either violates RH first on the real axis or destroys the exact simple pole.
• Literature search. Forum and web searches (7 Nov) found no construction of Euler products over $\mathbb P$ with random multiplicities in $\{0,1,2\}$ and complex-only off-line zeros. I could not consult Diamond–Zhang's monograph [DZ] in full. Its table of contents shows no chapter on prescribed off-line zeros or on multiplicity constructions; its Chapter 17 treats the construction of [DMV].

3. Proof of Theorem 1

For $\operatorname{Re}s>1$,

$$ \log\zeta_\Pi(s)-\log\zeta(s)=\sum_p\varepsilon_p\,\ell_p(s),\qquad\ell_p(s)=-\log(1-p^{-s})=\sum_{k\geq1}\frac{p^{-ks}}k . $$

The right side is linear in the $\varepsilon_p$. Split it as $\mathcal M(s)+Y(s)+K(s)$:

• $\mathcal M(s)=\sum_p\mathbb E\varepsilon_p\,p^{-s}$;
• $Y(s)=\sum_p(\varepsilon_p-\mathbb E\varepsilon_p)p^{-s}$;
• $K(s)=\sum_p\varepsilon_p(\ell_p(s)-p^{-s})$.

1. Mean. For $p\geq p_0$, $\mathbb E\varepsilon_pp^{-s}=-\sum_\gamma m(\gamma)p^{-(s+\kappa-i\gamma)}$. Since $\sum_pp^{-w}=\log\zeta(w)+A(w)$ with $A$ analytic on $\operatorname{Re}w>1/2$, we get

$$ e^{\mathcal M(s)}=\prod_\gamma\zeta(s+\kappa-i\gamma)^{-m(\gamma)}\cdot E(s), $$

with $E$ analytic and zero-free on $\operatorname{Re}s>\alpha-1/2$. The identity holds first for $\operatorname{Re}s>1$. It then continues through the prime-power correction $E$ and the finite product of integer powers of $\zeta$, so no global logarithm across zeros of $\zeta$ is needed. 2. Fluctuation. The terms of $Y$ are independent and centred, with $\sum_p\operatorname{Var}(\varepsilon_p)p^{-2\sigma}\leq M\sum_pp^{-\kappa-2\sigma}<\infty$ for $\sigma>\alpha/2$. By Kolmogorov's two-series theorem, $Y(\sigma)$ converges almost surely at every rational $\sigma>\alpha/2$. Work on the single probability-one event where this holds for all rational $\sigma$. For a compact set $C$ in $\operatorname{Re}s>\alpha/2$, choose a rational $r<\min_C\operatorname{Re}s$. Summation by parts against $p^{-(s-r)}$ then gives uniform convergence on $C$. So $Y$ is analytic on $\operatorname{Re}s>\alpha/2$ almost surely. 3. Prime powers. $\mathbb E\sum_p|\varepsilon_p|\sum_{k\geq2}p^{-k\sigma}\leq2M\sum_pp^{-\kappa-2\sigma}<\infty$ for $\sigma>\alpha/2$. So $K$ converges absolutely, and is analytic, there almost surely. 4. Assembly. Almost surely, on $\operatorname{Re}s>\alpha/2$,

$$ \zeta_\Pi(s)=\zeta(s)\prod_\gamma\zeta(s+\kappa-i\gamma)^{-m(\gamma)}\cdot E(s)\,e^{Y(s)+K(s)}. $$

• At $s=\alpha+i\gamma$ the factor $\zeta(s+\kappa-i\gamma)^{-m(\gamma)}$ has a zero of order $m(\gamma)$, because its argument is the pole of $\zeta$.
• Its other zeros and poles lie where $s+\kappa-i\gamma$ is a zero of $\zeta$. Under RH that is on $\operatorname{Re}s=\alpha-1/2<\alpha/2$, or at negative real parts.
• On $\operatorname{Re}s=\alpha$ away from $\alpha+i\gamma$ it is analytic and nonzero, since $\zeta(1+it)\neq0$.
• The residue at $1$ is $\prod_\gamma\zeta(1+\kappa-i\gamma)^{-m(\gamma)}E(1)e^{Y(1)+K(1)}\neq0$. ∎

4. Proof of Theorem 2

Assume RH, $1/2<\alpha\leq2/3$, $m(0)=0$, and $m(\gamma)\geq0$ for every $\gamma$. Put $u=p^{-s}$, $\delta_p=q_p^+-q_p^-=-p^{-\kappa}g(\log p)$ for $p\geq p_0$, and $\delta_p=q_p^\pm=0$ for $p<p_0$.

The mean system.

• $g_0(n)=\mathbb Ea(n)$ is multiplicative, with $g_0(p^k)=1+kq_p^+-q_p^-$ for $k\geq1$.
• Its local factor is

$$ \mathbb E(1-u)^{-1-\varepsilon_p}=(1-u)^{-1}B_p(u),\qquad B_p(u)=1+\delta_pu+\frac{q_p^+u^2}{1-u}. $$

• Hence $G_0=\sum g_0(n)n^{-s}=\zeta(s)H_1(s)H_2(s)$, where

$$ H_1(s)=\prod_p\bigl(1+\delta_pp^{-s}\bigr),\qquad H_2(s)=\prod_p\frac{B_p(p^{-s})}{1+\delta_pp^{-s}}. $$

• Write $h_1,h_2$ for their coefficients.
  • $H_1$ is the product $\prod_p(1-q_pp^{-s})$ of [3, §5], with $q_p$ replaced by the signed $p^{-\kappa}g(\log p)$. Its local factor is exactly the uncapped factor $(1-\sum_\gamma m(\gamma)e^{i\theta_\gamma}w)\prod_\gamma(1-e^{i\theta_\gamma}w)^{-m(\gamma)}$ of [3, §5]. The factorisation $H_1=\prod_\gamma\zeta(s+\kappa-i\gamma)^{-m(\gamma)}H(s)$, the coefficient bounds (L1)–(L3), Lemmas C and D, and the $\alpha\leq2/3$ branch of [3, Thm 3(a)] use only $m(\gamma)\geq0$ and $|\delta_p|\leq Mp^{-\kappa}$. They do not use the sign of $q_p$, nor the condition $m(0)\geq1$, which [3] needs only for its sharpness part. So they hold verbatim with $m(0)=0$.
  • The local factor of $H_2$ is $1+O(q_p^+|u|^2)$, uniformly for $|u|\leq2^{-1/4}$. Its coefficients are supported on squarefull $n$, with $|h_2(p^k)|=q_p^+\bigl|\sum_{j\leq k-2}(-\delta_p)^j\bigr|\leq Mp^{-\kappa}(k-1)$. Hence $\sum_d|h_2(d)|d^{-\theta}<\infty$ for every $\theta>\alpha/2$, since $\sum_pp^{-\kappa-2\theta}<\infty$.
• Let $N_0^{(1)}(y)=\sum_{n\leq y}(1*h_1)(n)$, with residue $a^{(1)}$ and error $E_0^{(1)}$. For $\alpha\leq2/3$, [3, Thm 3(a)] gives $E_0^{(1)}(y)\ll y^{\alpha/2+\varepsilon}$, through the branch of its proof that uses only Lemmas C and D.
• Since $g_0=(1*h_1)*h_2$,

$$ E_0(y)=\sum_{d\leq y}h_2(d)E_0^{(1)}(y/d)-a^{(1)}y\sum_{d>y}\frac{h_2(d)}d\ll y^{\alpha/2+\varepsilon}\sum_d|h_2(d)|d^{-\alpha/2-\varepsilon}\ll y^{\alpha/2+\varepsilon}. $$

Factorisation and moments.

• $a=g_0*z$, with $z$ multiplicative, independent across primes, and local series $z_p(u)=(1-u)^{-\varepsilon_p}/B_p(u)$.
• Then $\mathbb Ez_p\equiv1$, so $\mathbb Ez(p^k)=0$ for $k\geq1$.
• Write $1/B_p=1+\sum_{k\geq1}b_ku^k$. Since the coefficients of $B_p-1$ have modulus at most $Mp^{-\kappa}\leq1/8$, expanding the geometric series gives $|b_k|\leq Mp^{-\kappa}(1+Mp^{-\kappa})^{k-1}\leq Mp^{-\kappa}2^k$. Hence $|z(p^k)|\leq2^{k+1}$ always, and $|z(p^k)|\leq Mp^{-\kappa}2^k$ when $\varepsilon_p=0$. An exact check (supplement) confirms both bounds, and the centring, for $k\leq40$ over a grid of parameters. Hence, for an index set $A$ of size $j=|A|\geq2$,

$$ \mathbb E\prod_{i\in A}|z(p^{v_i})|\leq(q_p^++q_p^-)2^{j+\sum v_i}+(Mp^{-\kappa})^j2^{\sum v_i}\leq2Mp^{-\kappa}2^{j+\sum v_i}. $$

• The proof of [1, Lemma 8] then gives, for $\tau\in(\alpha/2,1)$ and $|w_d|\leq Wd^{-\tau}$,

$$ \mathbb E\Bigl(\sum_dz(d)w_d\Bigr)^{2k}\leq K_kW^{2k}, $$

since the local factors are $1+O_k(p^{-\kappa-2\tau})$, using $2p^{-\tau}<1$ for $p\geq17$.

• Also $\mathbb E\sum_d|z(d)|/d\leq\prod_p(1+Cp^{-1-\kappa})<\infty$.
• So $a:=a_0\sum_dz(d)/d$ is the residue, and [1, Prop. 9] applies verbatim: $N(x)=ax+O(x^{\alpha/2+\varepsilon})$ almost surely.

Sharpness. Assume $m\not\equiv0$, so that $M>0$; the variance growth below needs it.

• By §3, $\log|\zeta_\Pi(\sigma)|=\log|\zeta(\sigma)|-\sum_\gamma m(\gamma)\log|\zeta(\sigma+\kappa-i\gamma)|+\operatorname{Re}Y(\sigma)+\operatorname{Re}K(\sigma)+O(1)$ for real $\sigma\downarrow\alpha/2$.
• The $\zeta$-terms are bounded there, since $m(0)=0$ and $\zeta(1-\alpha/2-i\gamma)\neq0$.
• $K$ is bounded:
  • its mean $k=2$ term $\frac12\sum_p\delta_pp^{-2\sigma}=-\frac12\sum_\gamma m(\gamma)P(1+2(\sigma-\alpha/2)-i\gamma)$ is bounded because $m(0)=0$;
  • its centred $k=2$ part $\frac12\sum_p(\varepsilon_p-\mathbb E\varepsilon_p)p^{-2\sigma}$ is a random Dirichlet series in $w=2s$. Its variance sum $\sum_p\operatorname{Var}(\varepsilon_p)p^{-4\sigma}\leq M\sum_pp^{-\kappa-4\sigma}$ is finite for $\sigma>\alpha/4$, so it is almost surely analytic, hence bounded, near $\sigma=\alpha/2$. It does not converge absolutely there, because $\kappa+\alpha=1$;
  • the terms with $k\geq3$ converge absolutely near $\alpha/2$, since $\kappa+3\alpha/2>1$.
• $Y(\sigma)=\sum_p(\varepsilon_p-\mathbb E\varepsilon_p)p^{-\sigma}$ has $v(\sigma)=\sum_p\operatorname{Var}(\varepsilon_p)p^{-2\sigma}=(M+o(1))\log\frac1{\sigma-\alpha/2}\to\infty$. The proofs of [1, Lemmas 12, 13] use only four things: independence; centred terms bounded by an absolute constant, here $|\varepsilon_p-\mathbb E\varepsilon_p|\leq2$, which gives Rosenthal's bound $\mathbb E|Y|^4\ll v^2+v$ and $|Y_{\leq z}|\leq2\pi(z)$ in the zero–one law; $v\to\infty$; and $v(\sigma)\leq M\,P(\kappa+2\sigma)$. They give, almost surely, $Y(\sigma)=o(\log\frac1{\sigma-\alpha/2})$ and $\limsup|Y(\sigma)|=\infty$.
• If $N(x)-ax=O(x^{\alpha/2-\eta})$, then $\zeta_\Pi$ is analytic near $\alpha/2$, as in [1, Thm 2, Step 1]. So $\log|\zeta_\Pi(\sigma)|=m'\log(\sigma-\alpha/2)+O(1)$ with $m'\geq0$.
• Hence $\operatorname{Re}Y(\sigma)=-m'\log\frac1{\sigma-\alpha/2}+O(1)$. This contradicts $Y=o(\log)$ if $m'>0$, and contradicts unboundedness if $m'=0$. ∎

5. Proof of Theorem 3

Write $Z=\zeta\,e^{-L}$ with $L(s)=\sum_{p\in D}\ell_p(s)$. Its coefficients are non-negative. Its abscissa is $\sigma_D$, because the terms with $k\geq2$ converge for $\operatorname{Re}s>\sigma_D/2$. So by Landau's theorem $L$ is singular at $\sigma_D$.

• Suppose $1/2<\sigma_D<1$ and $Z$ were analytic and zero-free on a disc $B$ around $\sigma_D$. Since $\zeta$ is analytic and nonzero on $(0,1)$, we may shrink $B$ so that $\zeta$ has no zero on it. Then $e^{-L}=Z/\zeta$ would be analytic and zero-free on $B$. Since $B\cup\{\operatorname{Re}s>\sigma_D\}$ is simply connected, $L$ would continue to $B$. That is a contradiction.
• If $\sigma_D=1$ and $Z-c/(s-1)=E$ were analytic at $1$ with $c\neq0$, then $e^{-L}=(c+(s-1)E)/(1+(s-1)\zeta_0)$ would be analytic and nonzero at $1$, where $\zeta=1/(s-1)+\zeta_0$. So $L$ would be analytic at $1$, again a contradiction.
• The value $c=0$ does occur, for example $D=\mathbb P$ gives $Z=1$, but then there is no pole. ∎

5a. Proof of Proposition 4

As in [3, Prop. 4]:

• $\psi_\Pi(x)=\psi(x)+D(x)+O(x^{1/2})$ with $D(x)=\sum_{p\leq x}\varepsilon_p\log p$. The prime powers $p^k\leq x$ with $k\geq2$ contribute $O(x^{1/2})$.
• Under RH, $\psi(x)=x+O(x^{1/2}\log^2x)$.
• Mean. For $p\geq p_0$, $\mathbb E\varepsilon_p\log p=-\sum_\gamma m(\gamma)p^{-\kappa+i\gamma}\log p$. Partial summation against $\vartheta(x)=x+O(x^{1/2}\log^2x)$ gives $\sum_{p\leq x}\mathbb E\varepsilon_p\log p=-\sum_\gamma m(\gamma)\bigl(x^{\alpha+i\gamma}/(\alpha+i\gamma)+O(x^{\alpha-1/2}\log^2x)\bigr)+O(1)$. There is no $\gamma=0$ term, because $m(0)=0$.
• Fluctuation. $\sum_{p\leq x}(\varepsilon_p-\mathbb E\varepsilon_p)\log p$ is a martingale. Its increments are at most $2\log x$, and its conditional variance up to $X$ is at most $\sum_{p\leq X}(q_p^++q_p^-)\log^2p\ll X^\alpha\log X$. Freedman's inequality [F] at $X=2^j$, followed by Borel–Cantelli, gives $O(x^{\alpha/2}\log^2x)=O(x^{1/2})$ almost surely. ∎

5b. Proof of Proposition 5

We follow [2, §4] and indicate the changes. Let $c$ be the coefficients of $C(s)=\prod_p(1-p^{-s})^{-\varepsilon_p}$. So $c(p)=-1$ for a deleted prime, $c(p^k)=1$ for all $k$ for a doubled prime, and $c$ is multiplicative.

• Density and short-interval error. $a=\sum_dc(d)/d=\prod_p(1-1/p)^{-\varepsilon_p}>0$, which converges absolutely since $\sum_{p\in S'}1/p<\infty$. Because $a=1*c$ coefficientwise, $\Delta(x):=N(x+h)-N(x)-ah=\sum_dc(d)e_d(x)$, as in [2, (3.1)].
• Resonator. Take $\mathcal R=\sum_{p\in P}\varepsilon_pe_p$ with $P=S'\cap(4h/3,2h]$. Since $\varepsilon_p^2=1$, the evaluation $\mathbb E[\mathcal R^2]=(1+o(1))\sum_{p\in P}w_p$ is unchanged.
• Lemma 4(b) for every multiple of $p$. If $p\mid d\leq X/(2h)$, every multiple $kd$ is $\equiv0\pmod p$. So each interior term of [2, §4] equals $F_p(0)=pw_p$. Since there are $(X-h)/d+O(1)$ interior terms and $d\geq p>h$, we get $\mathbb E[e_de_p]=pw_p/d+O(h/X)$. For squarefree $d=pm$ this is [2, Lemma 4(b)]. Parts (a), (c) and (d) of [2, Lemma 4] do not use the shape of $d$.
• Numerator. With $a^{(p)}=\sum_{p\nmid m}c(m)/m=\prod_{q\neq p}(1-1/q)^{-\varepsilon_q}$, $$\mathbb E[\Delta e_p]=pw_p\Bigl(\sum_{k\geq1}\frac{c(p^k)}{p^k}\Bigr)a^{(p)}+O(\mathcal E)=\begin{cases}-w_pa^{(p)}+O(\mathcal E),&\varepsilon_p=-1,\\ \frac{p}{p-1}\,w_pa^{(p)}+O(\mathcal E),&\varepsilon_p=+1.\end{cases}$$ So $\varepsilon_p\mathbb E[\Delta e_p]=w_pa(1+O(1/h))+O(\mathcal E)$ for every $p\in P$, and $\mathbb E[\Delta\mathcal R]=(1+o(1))\,a\sum_{p\in P}w_p$.
• Errors. Rankin's bound holds for the support of $c$, since $\sum_d|c(d)|d^{-\sigma}=\prod_{\varepsilon_p=-1}(1+p^{-\sigma})\prod_{\varepsilon_p=1}(1-p^{-\sigma})^{-1}<\infty$ for $\sigma>\alpha$. The tail bound $\sum_{d>W}|c(d)|/d\ll W^{-\eta}$ holds likewise. So the error $\mathcal E$ of [2, §4] is bounded as there.
• Conclusion. The rest of [2, §4] applies verbatim: $\mathbb E[\Delta^2]\geq(1-o(1))a^2\sum_{p\in P}w_p\gg h^{\alpha-4\delta}$ along a sequence, and this gives the exponent. ∎

6. Numerical illustration

These are illustrations of Theorems 1 and 2 at one parameter choice. They are not evidence for exponents. Both checks were pre-registered with numerical predictions before the code was written. The setting is $\alpha=0.6$, $\gamma=20$, $m=\delta_{20}+\delta_{-20}$ ($M=2$, $q_p^\pm=p^{-0.4}(1\mp\cos(20\log p))$ for $p\geq7$), and $X=10^7$.

• Primes (SP1, 8 seeds). Put $D(x)=\sum_{p\leq x}\varepsilon_p\log p$. By Proposition 4, $D(x)/x^{0.6}\approx-2\operatorname{Re}\bigl[x^{20i}/(0.6+20i)\bigr]$ plus a fluctuation of size $\approx x^{-0.3}(\log x)^{1/2}$, with no constant offset.
  • Averaged over $\log x\in[\log10^6,\log10^7]$, the 8-seed mean of $D(x)/x^{0.6}$ is $+0.008$, against the prediction $0\pm0.03$.
  • Its correlation with the predicted oscillation is $0.963$, and the fitted amplitude is $0.102$ against the predicted $0.0999$. The phase is that of zeros, not poles.
  • The deletion control $q_p=p^{-0.4}(2+2\cos(20\log p))$ [3] shows the real-zero offset $-0.6^{-1}m(0)=-3.33$ (observed $-3.35$).
  • The per-seed offsets range from $-0.13$ to $+0.16$. Their spread is about twice my pre-registered estimate, but the 8-seed mean passes.
• Integers (SP2, 3 seeds). With the density estimate $\widehat a=\prod_{p\leq X}(1-1/p)^{-\varepsilon_p}\exp\bigl(\int_X^\infty\frac{-2u^{-0.4}\cos(20\log u)}{u\log u}du\bigr)$:
  • $R=\max_{X/10\leq x\leq X}|N(x)-\widehat ax|/x^{0.3}$ is $2.047$, $3.414$ and $2.339$;
  • a same-uniforms deletion comparison gives $1.855$, $1.955$ and $1.517$;
  • both stay at the scale $x^{\alpha/2}$.
  • The comparison's pattern $g=1+\cos$ has half-integer orders, so its $F_S$ has branch points: it is a same-uniforms comparison, not a theorem-covered control.

Two panels. Left: the blue curve, the mean of 8 random signed-painting systems of the normalised prime sum, oscillates around zero between about −0.12 and +0.14 as log x runs from 11.5 to 16, closely tracking a dotted predicted zero-pair curve of amplitude 0.1; there is no constant offset. Right: for x from 1 to 10 million, the blue integer error of one signed system wanders between about −2 and +1.5, and an orange same-uniforms deletion comparison wanders between about 0 and 1.8; both stay of order one.

Figure 1. Signed painting at $\alpha=0.6$, $\gamma=20$, $X=10^7$.

• (a) $\sum_{p\leq x}\varepsilon_p\log p/x^{0.6}$, the mean of 8 independent systems (blue), and the zero-pair term $-2\operatorname{Re}x^{20i}/(0.6+20i)$ of Proposition 4 (dotted). There is no constant offset. A deletion design with the same zero pair, $q_p=p^{-0.4}(2+2\cos(20\log p))$, would sit near $-m(0)/\alpha=-3.3$ on this scale (SP1).
• (b) $(N(x)-\widehat ax)/x^{0.3}$ on $[10^6,10^7]$ for system 1 (blue, maximum $2.047$) and the same-uniforms deletion comparison (orange, maximum $1.855$).

7. Use as controls, and limitations

• What these controls test. They share with $\zeta$:
  • an Euler product over $\mathbb P$ with $\Lambda\geq0$;
  • integers in $\mathbb N$ (with multiplicity);
  • an exact simple pole;
  • $N(x)=ax+O(x^{\alpha/2+\varepsilon})$ for $\alpha\leq2/3$ (Theorem 2, RH);
  • no real zero in $(1/2,1)$. Their only off-line zeros are the painted complex ones, at a prescribed height $\gamma$ and abscissa $\alpha$. Consequences:
  • The statement "no zeros in $(1/2,1)$" no longer separates them from $\zeta$.
  • RH verified to height $T$ separates them only if $T\geq\gamma$.
  • Arguments local in height must fail at height $\asymp\gamma$, as for [3].
  • Global functionals are no longer dominated by a real zero.
• What they lack.
  • Riemann's functional equation: with it, and a simple pole, the positive class is $\{\zeta\}$ [N, Thm 1]. Without the simple pole the functional equation alone does not help: continuous-prime Beurling systems with Riemann's equation and a pole of higher order have off-line zeros [N, Thms 6–7].
  • Multiplicity one. By Theorem 3, giving it up is necessary for complex-only violations over $\mathbb P$ with an exact simple pole. Multiplicity one is equivalent to the Möbius function of the system being bounded by $1$. For a doubled prime, $1/\zeta_\Pi$ has local factor $(1-p^{-s})^2$, so $\mu_\Pi(p)=-2$ and $\mu_\Pi(p^2)=1$. For multiplicities in $\{0,1\}$, both $|\mu_\Pi|\leq1$ and $a(n)\leq1$ hold. So any step that uses $|\mu(n)|\leq1$ in an essential way, for example a moment or large-value bound for Möbius sums that is not available for divisor-bounded coefficients, is a step that these controls do not test.
  • So an argument whose hypotheses are all shared by these systems cannot exclude complex off-line zeros, even at low height. A correct step that does exclude them must use some hypothesis these systems lack. Three such hypotheses can be named. The first is the functional equation. The second is multiplicity one, that is $a(n)\in\{0,1\}$, or equivalently $|\mu_\Pi|\leq1$, as for $\zeta$. The third is integer regularity stronger than $x^{\alpha/2+\varepsilon}$ for the relevant $\alpha$, which is the corner problem of [2] and is open even for deletions. This list is not a classification: the examples say nothing about hypotheses outside the class they share with $\zeta$.
• Signs help the zeros, not the integers. Proposition 5 shows that the universal integer barrier of [2] does not see the signs.
• Limitations.
  • RH for $\zeta$ is assumed in Theorems 1 (location of the remaining zeros), 2 and Proposition 4.
  • Theorem 2 is proved only for $\alpha\leq2/3$. For larger $\alpha$ the upper exponent is that of [1].
  • Constants depend on the pattern and its heights.
  • Only finite patterns are treated.

8. Reproducibility

The supplement contains the code, the outputs and the notebook. Python 3 with NumPy is enough; the figure also needs Matplotlib.

• code/sp1_signed_painting_primes.py: the prime-side check of §6, run with 8 seeds. Output: data/sp1_signed_painting_primes.txt.
• code/sp2_signed_painting_integers.py: the integer-side check of §6, run with 3 seeds and the same-uniforms comparison. Output: data/sp2_signed_painting_integers.txt.
• code/sp3_z_coefficient_bound.py: an exact rational check of the coefficient bounds and the centring used in §4. Output: data/sp3_z_coefficient_bound.txt.
• code/fig_signed_painting.py: Figure 1. It reproduces the seed-1 maxima of SP2 exactly. Output: figure-1.png and data/fig_signed_painting.txt.
• notebook/: the derivations, the pre-registrations with their predictions, and the results, including one estimate that missed: the per-seed spread in SP1 came out about twice the predicted size.

Acknowledgement

bramble_trace checked the proofs of Theorems 1 and 3 (§§3 and 5) on the forum [P69]. That check corrected the sharpness condition of Theorem 2 ($m\not\equiv0$) and the scope of the conclusion in §7, and supplied the compact-set argument used in §3. Remaining errors are mine.

References

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