bibkey: revesz2022prescribedzeros
authors: Szilard Gy. Revesz
year: 2022
title: Oscillation of the remainder term in the prime number theorem of Beurling, caused by a given zeta-zero
doi: null
url: https://arxiv.org/abs/2202.01837v3
claim: Theorem 8 realizes prescribed zeros in a discrete Beurling system. Its application below tests the joint Robin-source constraints with full prime powers and reserve, without an ordinary-integer bound.
strata_touched: []
license: citation-only
triage: anchor

Prescribed Beurling zeros and the joint Robin-source boundary

The inspected primary is arXiv:2202.01837v3, whose arXiv version stamp is 15 September 2022. The PDF has 28 pages and SHA-256 513da18b5e8c9054f5cce499ae5c32340aff60640b759b1ddd3a66eaee772f89. Section 7, printed pp.16–22, especially equations (50)–(51), their Chebyshev consequence on p.17, and Theorem 8 on pp.21–22, were read. The complete source proof and a journal-version correspondence have not been independently audited. No source text is vendored or Lean/kernel verification claimed.

Reuse the prescribed-zero construction

Theorem 8 takes $1/2\le r<1$ and a finite conjugation-symmetric multiset $\mathcal{S}$ with $r<\Re \rho <1$. At $r=1/2$, choose once and for all

$$\mathcal{S}=\{\rho ,\bar{\rho }\},\rho =\beta +i\tau ,1/2<\beta <1,\tau >0.$$

The frequency $\tau$ is fixed; it may exceed any specified finite height. It supplies one discrete Beurling system, with prime generators of norms $p>1$, for which, for every fixed $\epsilon >0$,

$$\mathcal{N}(x)=\kappa x+O_{\epsilon }(x^{1/2+\epsilon }),{\psi }_{\mathcal{P}}(x)-x=-2\Re \frac{x^{\rho }}{\rho }+O(\sqrt{x}).\text{\hspace{1em}(RJ1)}$$

Here $\kappa >0$. The source’s intermediate error $\sqrt{x}\exp (O((\log x{)}^{2/3}))$ is $O_{\epsilon }(x^{1/2+\epsilon })$, because $(\log x{)}^{2/3}=o(\log x)$.

Its construction also supplies ${\theta }_{\mathcal{P}}(x)=f_0(x)+O(\log x)$, with $f_0^{\prime }(x)=1+o(1)$. This local information is read from (50)–(51) and the p.17 calculation; it is stronger than the $O(\sqrt{x})$ error alone. The source explicitly uses the published Broucke–Vindas realization. Neither that construction nor the prescribed-zero theorem is re-proved.

A generalized integer here is a finite exponent vector in the free commutative semigroup on those prime generators. Its norm is $|\mathfrak{n}|=\prod p^{a_p}$. Distinct vectors remain distinct even when their real norms coincide. All counts retain that multiplicity. Define

$$Z_{\mathcal{P}}(\mathfrak{n})=\prod _p\frac{1-p^{-a_p-1}}{1-p^{-1}},G_{\mathcal{P}}(\mathfrak{n})=\frac{Z_{\mathcal{P}}(\mathfrak{n})}{\log \log |\mathfrak{n}|}(|\mathfrak{n}|>e).$$

The application below concerns these actual vectors, not independently chosen support and exponent bounds. Its additional argument transports the prescribed oscillation to their joint extremal sources. This is a paper application, without a literature-priority or formal-certification claim.

Normalize once while preserving any finite ordinary prefix

Fix any $D\ge 2$. Replace the finitely many prime generators of norm at most $D$ by exactly one generator for each ordinary prime at most $D$. Partial summation of ${\theta }_{\mathcal{P}}(x)\sim x$ gives ${\pi }_{\mathcal{P}}(x)\sim x/\log x$, so ${\sum }_{p>D}{p}^{-1}=\infty$ even after this finite replacement. Delete finitely many further generators of norm greater than $D$ so that the residue ${\kappa }^{\prime }$ lies in $(0,1)$. Choose an integer $m\ge 1$ large enough that

$$q=(1-({\kappa }^{\prime }{)}^{1/m}{)}^{-1}>D,$$

and add $m$ distinct generators, each of norm $q$. The new residue is exactly ${\kappa }^{\prime }/(1-q^{-1}{)}^m=1$. No generator of norm at most $D$ is changed by these last two steps.

A finite change of generators changes ${\psi }_{\mathcal{P}}(x)$ by $O(\log x)$ and eventually changes ${\theta }_{\mathcal{P}}(x)$ by a constant. Finite inclusion–exclusion handles deletion. For each insertion, ${\mathcal{N}}_{\mathrm{new}}(x)={\sum }_{q^k\le x}{\mathcal{N}}_{\mathrm{old}}(x/q^k)$ and its error is a convergent geometric sum. Iterating this finite number of times gives

$$\mathcal{N}(x)=x+O_{\epsilon }(x^{1/2+\epsilon }).\text{\hspace{1em}(RJ2)}$$

Thus (RJ1) and the local $\theta$ property survive. All following sources belong to this one fixed normalized system. Added generators remain distinct labels when their norms coincide. Its prime prefix, generalized integers, and divisor weights up to $D$ now agree exactly with the ordinary ones; in particular $\mathcal{N}(x)=\lfloor x\rfloor$ for $1\le x\le D$.

The finite Euler factors have zeros and poles only on $\Re s=0$. Thus the prescribed right-half-plane zeros are unchanged. Choosing $\tau >H$ for any fixed $H$ also preserves absence of zeros in $\Re s>1/2$, $|\Im s|\le H$. This does not prescribe the ordinary critical-line zero table. All constants may depend on the fixed $D,H,\beta ,\tau$ and finite modifications; no uniformity as these grow is asserted.

Keep the actual CA pressure and its complete reserve

For $t>1$, put $g(t)=1/(t\log t)$ and $w(t)=-g^{\prime }(t)=(1+\log t)/(t^2{\log }^{2}t)$. Define the actual pressure and an auxiliary pressure by

$$\mathcal{F}(ϵ)=\underset{\mathfrak{n}}{\max }\{\log Z_{\mathcal{P}}(\mathfrak{n})-ϵ\log |\mathfrak{n}|\},{\mathcal{F}}_0(ϵ)=\sum _{p,k\ge 1}{\left(\frac{1}{k{p}^k}-ϵ\log p\right)}_{+}.$$

Both optimizations have finite support at every positive price. The actual successive local gains are $d_{p,k}=\log (1+1/(p+\cdots +p^k))$, strictly decreasing in $k$. The auxiliary gain is selected at $ϵ=g(t)$ exactly when $p^k\le t$, with zero gain at equality. Hence, retaining inclusive prime powers,

$${\mathcal{F}}_0(g(t))=J(t)-g(t){\psi }_{\mathcal{P}}(t),J(t)=\sum _{p^k\le t}\frac{1}{k{p}^k}.$$

Set

$$P(t)=\gamma +\log \log t-\frac{1}{\log t}-\mathcal{F}(g(t)),R(t)={\mathcal{F}}_0(g(t))-\mathcal{F}(g(t)).\text{\hspace{1em}(RJ3)}$$

Here $R$ is the full difference of the actual optimized pressures. For $0<z<1$ and integers $a\ge 0$, the elementary comparison is

$$D_a(z)=\sum _{k=1}^a\frac{z^k}{k}-\log (1+z+\cdots +z^a),0\le D_a(z)\le -\log (1-z^{a+1}).$$

Indeed, $D_a^{\prime }(z)=z^a((a+1)/(1-z^{a+1})-1/(1-z))\ge 0$ and $D_a(0)=0$; the upper bound follows by summing the missing logarithmic tail. If $a_p$ is the actual optimal exponent and $b_p=\lfloor \log t/\log p\rfloor$, evaluating each local objective at the other’s maximizer gives

$$D_{a_p}(p^{-1})\le {\mathcal{F}}_{0,p}(g(t))-{\mathcal{F}}_p(g(t))\le D_{b_p}(p^{-1}).$$

For $p>t$, both exponents vanish. For $p\le \sqrt{t}$, $p^{b_p+1}>t$; for $\sqrt{t}<p\le t$, $b_p=1$. Using ${\pi }_{\mathcal{P}}(x)\ll x/\log x$ and ${\sum }_{p>y}{p}^{-2}\ll 1/(y\log y)$ therefore gives

$$0\le R(t)\le \sum _{p\le t}-\log (1-p^{-b_p-1})\ll \frac{1}{\sqrt{t}\log t}.\text{\hspace{1em}(RJ4)}$$

The constant in the Mertens expression must also be kept. By (RJ1), $J(t)-\log \log t\to C$. Abelian integration at $z\downarrow 0$ yields

$$\log {\zeta }_{\mathcal{P}}(1+z)=z{\int }_0^{\infty }{e}^{-zv}J(e^v)dv=-\log z+C-\gamma +o(1).$$

The normalized residue is one, so $C=\gamma$. The prime-power sums include atoms at their upper endpoint; at equality their auxiliary gain is zero. Stieltjes integration by parts with this convention gives the exact full-tail identities

$$\begin{array}{rl}I(t)& ={\int }_t^{\infty }({\psi }_{\mathcal{P}}(u)-u)w(u)du,\\ J(t)& =\log \log t+\gamma +g(t)({\psi }_{\mathcal{P}}(t)-t)-I(t),\\ P(t)& =I(t)+R(t).\end{array}\text{\hspace{1em}(RJ5)}$$

Let $B(t)$ be the logarithmic norm of the actual CA vector at price $g(t)$, with any maximizing convention at ties. Away from its locally finite events,

$$P^{\prime }(t)=(t-B(t))w(t),R^{\prime }(t)=({\psi }_{\mathcal{P}}(t)-B(t))w(t).$$

These pressures are continuous and locally absolutely continuous. Since $R(\infty )=0$, (RJ4) gives the complete reserve, including its future part,

$$R(t)={\int }_t^{\infty }(B(u)-{\psi }_{\mathcal{P}}(u))w(u)du.\text{\hspace{1em}(RJ6)}$$

The full real-clock inventory has the same normal form

The actual layer threshold $x_k(t)>1$ solves

$$\frac{\log (1+1/(x+x^2+\cdots +x^k))}{\log x}=g(t).$$

Expansion gives $x_1(t)=t+O(1)$ and $x_2(t)\sim \sqrt{2t}$. For a selected $k$th layer, $g(t)\log p\le d_{p,k}\le p^{-k}$. With $p_0>1$ the least prime norm, this implies $p^k\le H(t):=t\log t/\log p_0$ and $k=O(\log t)$. Consequently all layers $k\ge 3$ together are $O(t^{1/3}(\log t{)}^{4/3})=o(\sqrt{t})$.

The local property $\theta =f_0+O(\log x)$ with $f_0^{\prime }=1+o(1)$ controls bounded-length increments and individual jumps by $O(\log t)$. It therefore controls the $O(1)$ first-layer threshold shift and its ties. The second layer contributes $\sqrt{2t}+o(\sqrt{t})$. The inclusive prime-power identity similarly gives ${\psi }_{\mathcal{P}}(t)={\theta }_{\mathcal{P}}(t)+\sqrt{t}+o(\sqrt{t})$. Thus, on the entire real half-line and for every tie convention,

$$B(t)={\psi }_{\mathcal{P}}(t)+(\sqrt{2}-1)\sqrt{t}+o(\sqrt{t}).\text{\hspace{1em}(RJ7)}$$

Select one actual vector satisfying all joint source conditions

Integrating (RJ1) against the original kernel, without a finite cutoff, gives

$$I(t)=-2\Re \frac{t^{\rho -1}}{\rho (1-\rho )\log t}+O_{\rho }\left(\frac{t^{\beta -1}}{{\log }^{2}t}+\frac{1}{\sqrt{t}\log t}\right).\text{\hspace{1em}(RJ8)}$$

For example, substitute $v=\log u$ in ${\int }_t^{\infty }{u}^{\rho }w(u)du$ and integrate once by parts; the remainder is bounded by ${\int }_{\log t}^{\infty }{e}^{-(1-\beta )v}{v}^{-2}dv$. Together with (RJ4)–(RJ5), this yields, with fixed $K>0$ and $\varphi$,

$$P(t)=-K\frac{t^{\beta -1}}{\log t}\cos (\tau \log t+\varphi )+o(t^{\beta -1}/\log t),K=\frac{2}{|\rho (1-\rho )|},\varphi =\arg \frac{1}{\rho (1-\rho )}.$$

For each fixed source and parameters, the little-$o$ estimate holds on the whole tail: for every $ϵ>0$ its absolute remainder is at most $ϵt^{\beta -1}/\log t$ for all sufficiently large $t$. Choose $x_j=\exp (((2j+1)\pi -\varphi )/\tau )$, $d=e^{\pi /(3\tau )}$ and $y_j=e^{\pi /\tau }{x}_j$. Eventually $P>0$ on $[x_j,d{x}_j]$, whereas $P(y_j)\le -(K/2)y_j^{\beta -1}/\log y_j<0$. Since $P$ is continuous and tends to zero, it attains a negative minimum on $[x_j,\infty )$ at some $A_j>d{x}_j$. At an inventory event, $B$ jumps upward and $P^{\prime }$ jumps strictly downward, which rules out a local minimum there. Hence $A_j$ is event-free and $B(A_j)=A_j$. Its actual unique CA vector ${\mathfrak{N}}_j$ satisfies

$$\log |{\mathfrak{N}}_j|=A_j\to \infty ,P(A_j)=\gamma -\log G_{\mathcal{P}}({\mathfrak{N}}_j)=-h_j.\text{\hspace{1em}(RJ9)}$$

For every actual vector $\mathfrak{m}$ with $s=\log |\mathfrak{m}|>1$, the same pressure majorant gives $\log G_{\mathcal{P}}(\mathfrak{m})\le \gamma -P(s)$. The selected minimum consequently implies

$$G_{\mathcal{P}}(\mathfrak{m})\le G_{\mathcal{P}}({\mathfrak{N}}_j)\text{whenever }\log |\mathfrak{m}|\ge x_j.\text{\hspace{1em}(RJ10)}$$

This covers all future generalized integers, not only multiples. Every occupied prime has $p<A_j$, since $\log (1+p^{-1})/\log p<1/(p\log p)\le g(A_j)$ for $p\ge A_j$. Thus $\log |{\mathfrak{N}}_j/p|\ge A_j-\log A_j>x_j$ eventually, so (RJ10) also supplies every prime-deletion comparison. Moreover, $\Omega ({\mathfrak{N}}_j)\ge A_j/\log A_j\to \infty$: these are proper GA1 vectors in exactly the stated deletion sense.

If a prime generator $q\le A_j$ were absent, inserting it in the future comparison would force

$$1+q^{-1}\le \frac{\log (A_j+\log q)}{\log A_j}<1+\frac{\log q}{A_j\log A_j}\le 1+q^{-1},$$

a contradiction. The support is therefore the entire prime prefix to $A_j$. At the same source, all future real clocks satisfy

$${\int }_{A_j}^t(B(u)-u)w(u)du=P(A_j)-P(t)\le 0(t\ge A_j),A_j-{\theta }_{\mathcal{P}}(A_j)\sim \sqrt{2A_j}.\text{\hspace{1em}(RJ11)}$$

The original root-scale signed floor fails on those same sources

Write $a(t)=t^{\beta -1}/\log t$, eventually decreasing. The ratio $a(y_j)/a(d{x}_j)$ has a fixed positive lower bound, and $a(A_j)\le a(d{x}_j)$. The selected minimum therefore gives a fixed $c>0$ with $P(A_j)\le -ca(A_j)$. Nonnegativity of the complete reserve in (RJ4), with no independent reoptimization, yields

$$\sqrt{A_j}\log A_{j}I(A_j)\le -c{A}_j^{\beta -1/2}⟶-\infty .\text{\hspace{1em}(RJ12)}$$

Also $h_j\ge c{A}_j^{-(1-\beta )}/\log A_j$, which dominates $c_{\eta }{A}_j^{-\eta }$ for every fixed $1-\beta <\eta <1/2$ after adjusting a fixed positive constant. All prime powers, actual CA exponents, the full improper tail and the complete reserve remain in this same fixed system.

This tests the joint regular GA1 source obligation in a broader arithmetic category. Euler products, prime deletion and insertion, own-price CA selection, future-record comparisons and (RJ7) can coexist there with failure of the proposed root-scale floor. It is stronger source matching than an arbitrary pulse function or a first-prime-mass model, while reusing the classical realization machinery.

The conclusion does not establish insufficiency for ordinary integers: the finite-prefix matching and (RJ2) do not identify this system with ordinary arithmetic. Neither the exact global count $\lfloor x\rfloor$ nor the ordinary zeta functional equation is supplied by the construction. A proof for the original source must use an ordinary-integer or ordinary-zeta constraint that does not transport to this system. No such additional estimate, ordinary Robin counterexample, RH proof, new zero construction or Lean-certified theorem is supplied here.

The leading Selberg convolution relation is shared by this system

Tatuzawa–Iseki, On Selberg’s elementary proof of the prime-number theorem, Proc. Japan Acad. 27 (1951), 340–342, equation (2), p.340, records the ordinary relation

$$\psi (x)\log x+\sum _{n\le x}\Lambda (n)\psi (x/n)=2x\log x+O(x).$$

The three-page primary PDF and its equation (2) were inspected. This is a classical input, not a new prime-number theorem. The application below checks whether adding this particular relation distinguishes the already constructed system; it does not re-prove the realization or the source selection.

For a generalized integer vector $\mathfrak{a}$, let ${\Lambda }_{\mathcal{P}}(\mathfrak{a})=\log p$ when $\mathfrak{a}=p^k$, $k\ge 1$, and zero otherwise. Keep every prime power, generator label and ordered factorization, including coincident norms. Define

$$C_{\mathcal{P}}(x)=\sum _{|\mathfrak{ab}|\le x}{\Lambda }_{\mathcal{P}}(\mathfrak{a}){\Lambda }_{\mathcal{P}}(\mathfrak{b})=\sum _{|\mathfrak{a}|\le x}{\Lambda }_{\mathcal{P}}(\mathfrak{a}){\psi }_{\mathcal{P}}(x/|\mathfrak{a}|).$$

In the fixed normalized system of (RJ1), ${\psi }_{\mathcal{P}}(x)=x+O(x^{\beta })$ with $1/2<\beta <1$. The standard partial-summation consequences are

$$M(x):=\sum _{|\mathfrak{a}|\le x}\frac{{\Lambda }_{\mathcal{P}}(\mathfrak{a})}{|\mathfrak{a}|}=\log x+O(1),\sum _{|\mathfrak{a}|\le x}\frac{{\Lambda }_{\mathcal{P}}(\mathfrak{a})}{|\mathfrak{a}{|}^{\beta }}\ll x^{1-\beta }.$$

The constants can be enlarged so that the prime-error bound holds for all real arguments at least one. Thus the complete convolution gives

$$\begin{array}{rl}{C}_{\mathcal{P}}(x)& =xM(x)+O\left(x^{\beta }\sum _{|\mathfrak{a}|\le x}\frac{{\Lambda }_{\mathcal{P}}(\mathfrak{a})}{|\mathfrak{a}{|}^{\beta }}\right)=x\log x+O(x),\\ {\psi }_{\mathcal{P}}(x)\log x+C_{\mathcal{P}}(x)& =2x\log x+O(x).\end{array}$$

This retains the mixed convolution; no favorable sign was assigned to its error. The same system therefore satisfies the displayed leading Selberg relation while the original root-scale floor still fails on the actual sources (RJ9), as (RJ12) shows. Adding that relation to the already tested joint conditions does not supply the original root-scale floor. This tests the stated $O(x)$ relation, not every stronger weighted or source-dependent Selberg estimate.

The underlying exact convolution identity is also multiplicative rather than specific to the ordinary integer lattice. On $\Re s>1$, writing $D_{\mathcal{P}}=-{\zeta }_{\mathcal{P}}^{\prime }/{\zeta }_{\mathcal{P}}$ gives

$$-D_{\mathcal{P}}^{\prime }(s)+D_{\mathcal{P}}(s{)}^2=\frac{{\zeta }_{\mathcal{P}}^{\prime \prime }(s)}{{\zeta }_{\mathcal{P}}(s)}.$$

Its coefficients, indexed by generalized integer vectors, are $\Lambda_{\mathcal P}(\mathfrak a)\log|\mathfrak a| +(\Lambda_{\mathcal P}*\Lambda_{\mathcal P})(\mathfrak a)$. This identity holds in the same free multiplicative system, with no factorial or exact ordinary-counting assumption. Its logarithmically weighted coefficient sum differs from the first term of the displayed leading relation by the exact partial-summation identity

$${\psi }_{\mathcal{P}}(x)\log x-\sum _{|\mathfrak{a}|\le x}{\Lambda }_{\mathcal{P}}(\mathfrak{a})\log |\mathfrak{a}|={\int }_1^x\frac{{\psi }_{\mathcal{P}}(u)}{u}du=O(x).$$

Thus the two first terms are interchangeable at the stated $O(x)$ precision, not at an arbitrarily smaller error scale. Locally at a zero of multiplicity $m$, put $z=s-\rho$. Then $D_{\mathcal{P}}=-m/z+O(1)$ and the double-pole coefficient of $-D_{\mathcal{P}}^{\prime }+D_{\mathcal{P}}^2$ is $m(m-1)$. At a simple zero it cancels; a remaining simple pole depends on the actual numerator, so nonvanishing of ${\zeta }_{\mathcal{P}}^{\prime \prime }(\rho )$ is not assumed. These algebraic observations supply no remainder estimate.

The ordinary factorial identity uses additional global counting data; it is not identified with the universal convolution identity above. Deriving the ordinary floor from these tested conditions requires additional information not shared by this Beurling system. This application adds no new ordinary-prime estimate, realization theorem, RH proof or Lean/kernel certification, and makes no priority claim.