Prime additive excess

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Statement: proved

For every $M>0$ there is a finite singleton family of distinct prime moduli, with the original activation $n\ge q$, and a cutoff $X$, such that its delayed union $B$ and full periodic union measure $\beta$ satisfy

$$H_B(X)-\beta H_{\lfloor X\rfloor}>M. \tag{1}$$

Thus no finite constant depending only on $\sup\Omega(q)$ can bound this additive error: even the rank-one constant would be infinite. This supersedes the remaining small-exponential-rank-error target in rank and moment obstructions.

The proof amplifies the checked prime-template construction in finite sharp counterexamples. It uses that note's finite-complexity prime linear-forms theorem, not an unproved prime-tuple conjecture. The relevant source passages were checked: Green--Tao, Linear equations in primes, Lemma 1.6, printed p. 1761, and Main Theorem, p. 1763; and Green--Tao--Ziegler, An inverse theorem for the Gowers $U^{s+1}[N]$-norm, Theorem 1.3 and its following application paragraph, printed p. 1235. All systems of forms and all auxiliary parameters are fixed before the scale tends to infinity. No growing-complexity uniformity is used.

This is not a counterexample to Problem 25. Infinite prime-modulus systems have natural density by the pairwise-coprime theorem. The normalized excess in this construction is not bounded away from zero.

A uniform rough-number harmonic remainder

For an integer $y\ge2$, let

$$R_y=\{n\ge1:p\nmid n\text{ for all primes }p\le y\},\qquad F_y(x)=H_{R_y}(x),\qquad \delta_y=\prod_{p\le y}(1-1/p).$$

There is an absolute constant $C_0$ such that, for $x\ge1$,

$$0\le F_y(x)-\delta_y H_{\lfloor x\rfloor}\le C_0. \tag{2}$$

The elementary coprime upper sieve gives

$$|R_y\cap[1,t]|\le C\delta_y t\qquad(t\ge y). \tag{3}$$

Its theorem applies to every pairwise-coprime modulus set in $[2,\lfloor t\rfloor]$; there is no requirement that $t$ exceed a fixed power of $y$. Also $R_y\cap[1,y]=\{1\}$. Partial summation, splitting at $y$ when necessary, gives for $d\ge1$ and $x\ge y$

$$F_y(x)-F_y(x/d) \le C\delta_y(1+\log d)+\mathbf1_{d>x}. \tag{4}$$

If $x/d\ge y$, integrate (3) between $x/d$ and $x$. If $1\le x/d<y$, start at $y$, and use $\log(x/y)\le\log d$. If $d>x$, include the atom at one.

Let $D$ be a random positive $y$-smooth integer with $\Pr(D=d)=\delta_y/d$. The finite Euler product normalizes this law. Unique smooth/rough factorization yields

$$\mathbb E F_y(x/D)=\delta_y H_{\lfloor x\rfloor},\qquad \mathbb E\log D=\sum_{p\le y}\frac{\log p}{p-1}\ll\log(2y).$$

The Euler expansion gives $\delta_y\le1/H_y\ll1/\log(2y)$. Average (4) to obtain the upper bound in (2). Its lower bound follows from $D\ge1$ and monotonicity of $F_y$. For $1\le x<y$, $F_y(x)=1$ and $0\le\delta_y H_{\lfloor x\rfloor}\le1$, so (2) also holds there.

Prime-template amplification

Fix an arbitrary real $K>1$, put $\varepsilon=1/(4K)$, and choose a finite admissible set $S\subset\mathbb N$ with

$$L:=\sum_{t\in S}\log(1+1/t)\ge8K\log4.$$

The existence of such $S$ is proved in the prime-template note. Write $T=\max S$. These parameters are now fixed. Let $N$ tend to infinity through integers, and set

$$X=\lfloor N^K\rfloor.$$

Install the zero class at every prime at most $N$. For every good prime $q\in(N,X]$, independently retain it with probability $\varepsilon$, and then choose its residue uniformly among the prime-pattern shifts $a\in[0,q)$ for which every $a+tq$, $t\in S$, is prime. Here “good” has exactly the definition and moment estimates in the prime-template note. Let $A$ be the final delayed survivor set and $\delta$ its full Haar survivor measure.

The first and second prime moments proved in that note imply that the expected harmonic mass of deleted target primes in $((T+1)N,X]$ is at least

$$(1-e^{-\varepsilon L/2})\log K+o(1) \ge\tfrac34\log K+o(1). \tag{5}$$

Unlike its earlier application, this estimate allows every fixed $K>1$, not just $K<2$. The old restriction was used only to say that all old survivors were primes. For (5), it suffices that the target primes are old survivors, which remains true.

For completeness, the moment argument bounds target-prime survival by $e^{-\varepsilon J_{\rm good}(p)/2}$, where $J_{\rm good}\to L$ in prime-average $L^1$ on growing dyadic intervals. Partial summation then sums the errors to $o(1)$ between $N$ and $N^K$, with $K,S$ fixed. The boundary interval up to $(T+1)N$ has prime harmonic mass $o(1)$, and the full target mass is $\log K+o(1)$. Every used hit has $p=a+tq$ with $t\ge1$, so $p\ge q$: activation is exact. Thus

$$\mathbb E H_A(X)\le F_N(X)-\tfrac34\log K+o(1). \tag{6}$$

All moduli are distinct primes, so their periodic survivor measure is the product of their individual survivor fractions, independently of the chosen residues. Consequently

$$\begin{aligned} 0\le\delta_N-\mathbb E\delta &=\delta_N\left(1- \prod_{\substack{N<q\le X\\q\text{ good}}}(1-\varepsilon/q)\right)\\ &\le\varepsilon\delta_N\sum_{\substack{N<q\le X\\q\text{ prime}}}1/q. \end{aligned}$$

Since $\delta_N H_X\le H_X/H_N=K+o(1)$, prime reciprocal summation gives

$$(\delta_N-\mathbb E\delta)H_X \le\tfrac14\log K+o(1). \tag{7}$$

Combine (2), (6), and (7):

$$\begin{aligned} \mathbb E[\delta H_X-H_A(X)] &\ge-[F_N(X)-\delta_N H_X] +\tfrac12\log K+o(1)\\ &\ge\tfrac12\log K-C_0+o(1). \end{aligned} \tag{8}$$

Given $M$, first choose $K$ making the last main term greater than $M+1$; next fix $\varepsilon,S$; only then choose $N$ sufficiently large. Some realization has $\delta H_X-H_A(X)>M$, which is (1).

The finite moduli are singly assigned and can be ordered increasingly. There is no assertion that the realizations for different $M$ can be nested while retaining a positive Haar survivor measure. In fact their old prime sieves have $\delta_N\to0$. The constant normalized first-kill entropy bound and a uniform additive $o(\log X)$ bound are not contradicted by this result.

A related improvement of the preparatory-period restriction

Suppose a finite preparatory sieve has pairwise-coprime moduli, maximum modulus $N$, full survivor measure $\delta$, and period $P$. Append moduli $q>N$ coprime to $P$, all with one normalized integer center $0\le a<\min q$. The appended prepared-block estimates hold with an absolute constant even though $q$ need not exceed $P$:

$$h(X)^2\le C\tau,\qquad \sum_{i\text{ appended}}c_i(X)\log_+\frac{c_i(X)}{e_i} \le C\delta\log(C/\delta). \tag{9}$$

Here $h$ is total normalized harmonic appended first-kill mass and $\tau$ its total incremental Haar measure.

Indeed the same coprime upper sieve bounds the full old survivors below every $t\ge N$ by $C\delta t$. Every appended hit is beyond $N$, where all old classes are active. Partial summation gives $h(X)\le C\delta$. Independence gives $e_i=\delta e_i^0$ and $\tau=\delta\beta_0$, where the superscript and subscript refer to the appended zero-center system alone. Center subtraction and the valuation model used in prepared-block entropy give $c_i(X)\le C e_i^0$ and $h(X)\le C\beta_0$. These inequalities imply both parts of (9). If $\delta=0$, no appended first kills occur. This removes a period barrier for coprime preparations, not for arbitrary overlapping preparatory systems.