Li–Liu proof dependencies

For every \(A{\gt}0\) there are \(B\ge 0\), \(C{\gt}0\) and \(N_0\) such that \(\sum _{1\le q\le Q_B(N)}E^*(N,q)\le CN/(\log N)^A\) for integers \(N\ge N_0\), where \(Q_B(N)=\lfloor N^{1/2}/(\log N)^B\rfloor \) and \(E^*(N,q)=\max _{y\in \{ 0,\ldots ,N\} }\max _{0\le a{\lt}q,\, (a,q)=1}|\pi (y;q,a)-L_*(y)/\varphi (q)|\). Here \(L_*(x)=2/\log 2+\int _2^xdt/\log t\) for \(x{\gt}1\), while \(L_*(0)=L_*(1)=2/\log 2\) by the totalized interval-integral convention, not a principal value; \(E^*(N,0)=0\).

For each requested logarithmic saving, one common pair of Vaughan cutoffs pays the Type I, Type II and small-term means on the chosen high-conductor family, including the prefix and cofactor amplification factors.

Fix \(i,j,A\in \mathbb N\), \(C_{\rm scale}\ge 1\) and \(\zeta {\gt}0\). There is a threshold \(x_0\) depending only on these parameters such that for every real \(x\ge x_0\) the bound \(|\mathcal E_{\rm rect}|\le x/(\log x)^A\) holds uniformly in the following data: \(M,T\ge 1\), \(4MT=x\), \(T=x^\nu \), \(\zeta \le \nu \le 1/10+\zeta /10\); an integer residue \(0{\lt}N\le C_{\rm scale}x\); a finite set \(\mathcal U\subset [M,2M]\cap \mathbb N\) and interval \(J=(u,v]\cap \mathbb N\) with \(T\le u\le v\le 2T\); coefficients \(|\alpha (m)|\le \tau _i(m)\) on \(\mathcal U\), the short coefficient \(\beta _N(n)=\mathbf1_{n\ \text{ prime},\, (n,N)=1}\), and a signed well-factorable coefficient \(c\) of order \(j\) at \(Q=x^{(5-5\nu )/9-\zeta }\). Here \(\tau _i\) is the ordered \(i\)-fold divisor function, and \(\mathcal E_{\rm rect}\) is the signed sum over \(1\le q\le \lfloor Q\rfloor \), \((q,N)=1\), of \(c(q)\) times the \(\alpha \beta _N\) bilinear progression mass at \(mn\equiv N\pmod q\) minus its coprime mass divided by \(\varphi (q)\). The threshold precedes all rectangle, residue and coefficient choices.

For the constructed Jurkat–Richert functions \(F,f\), let \(c_\gamma =2e^\gamma \). On \(s{\gt}0\), define \(\widehat T^+(s)=s^{-2}\) for \(s\le 3\) and \([F(s)-f(s-1)]/(c_\gamma s)\) for \(s{\gt}3\); define \(\widehat T^-(s)=2s^{-2}\) for \(s\le 2\) and \([F(s-1)-f(s)]/(c_\gamma s)\) for \(s{\gt}2\). These positive extensions of the normalized derivatives \(-F'/c_\gamma \) and \(f'/c_\gamma \) satisfy the complete Suzuki source properties with \(\widehat\beta =2\): positivity, continuity, the initial formulas, weighted delay derivatives, and weighted and exponential decay.

\(\exists K{\gt}1:\ \prod _{p\in \mathcal P}(1-1/(p-1))^{-1}\le (\log z_2/\log z_1)(1+K/\log z_1)\) for every finite set of odd primes \(\mathcal P\subset [z_1,z_2)\), \(2\le z_1\le z_2\).

\(\exists C{\gt}0\ \forall n\ge 2:\ |\prod _{p\le n}(1-1/p)-e^{-\gamma }/\log n|\le C/(\log n)^2\).

\(\forall U{\gt}0\ \exists C{\gt}0,B\ge 0,N_0\ \forall N\ge N_0,I,a:\ \sum _{1\le q\le Q_B(N)}M_{a,I,*}(N;q)\le CN/(\log N)^U\), provided \(|a|\le 1\), \(A_2\le N^{2/3}\), \((\log N)^{2B}\le A_1\).

\(\exists c{\gt}0:\ \psi (x)-x=O(xe^{-c(\log x)^{1/10}})\).

Fix hats satisfying the Suzuki source properties and real \(d,\Delta ,\Theta \) with \(0{\lt}\Delta {\lt}1\), \(d{\gt}7/(1-\Delta )\), \(\Theta {\gt}0\), \(2/d{\lt}1/\Theta \) and \(2/\Theta +3/d{\lt}1-\Delta \). The source selects its auxiliary constants, and in particular \(C\ge 3\), before the bounding sieve \(S\) varies. For each \(K\ge 2\) satisfying its dimension-one local-product bound, each integer \(D\ge 2\) and \(4\le s\le 6\), put \(z=\lceil D^{1/s}\rceil \ge 2\) and assume \(s\le \sigma _{\rm src}(D;d)=(\log D)^{1/d}\log \log (27D)\). With \(\mathcal P_S(z)\) the supported primes below \(z\), \(P=\prod _{p\in \mathcal P_S(z)}p\), \(m=2(|\mathcal P_S(z)|+1)\) and \(V_S(z)=\prod _{p\in \mathcal P_S(z)}(1-g_S(p))\), one has \(V_S(z)[f(s)-C e^{\sqrt K}E_m(D,s;d)(\log D)^{-\Delta }]\le \sum _{e\mid P}\lambda _e^-g_S(e)\). The weights are the finite lower Rosser weights of natural level \(D\), and \(E_m(D,s;d)=(1+s^d/\log D)^s s\widehat T^{\eta _m}(s)\), with \(\eta _m=+\) for odd \(m\) and \(-\) for even \(m\).

Given hats satisfying the Suzuki source properties and fixed \(d,\Delta ,\Theta \) with \(0{\lt}\Delta {\lt}1\), \(d{\gt}7/(1-\Delta )\), \(\Theta {\gt}0\), \(2/d{\lt}1/\Theta \) and \(2/\Theta +3/d{\lt}1-\Delta \), for every \(K{\gt}1\) and \(\rho {\gt}0\) there is \(z_0\) before the bounding sieve \(S\) varies. If \(z\ge \max (2,z_0)\), \(D_{\rm real}{\gt}0\), \(S\) satisfies the dimension-one local-product bound with \(K\), every sifting prime is at most \(z\), and \(s=\log D_{\rm real}/\log z\in [3/2,4]\), then \(\sum _{e\mid P}\lambda _e^+g_S(e)\le (F(s)+\rho )\prod _{p\mid P}(1-g_S(p))\). Here \(P\) is the product of the sifting primes, the product on the right is over primes, and \(\lambda ^+\) is the finite upper Rosser weight at natural level \(\lfloor D_{\rm real}\rfloor +1\).

For every \(\delta {\gt}0\), there exists \(\varepsilon _0\in (0,2/15]\) such that, for every fixed \(0{\lt}\varepsilon {\lt}\varepsilon _0\), there is \(N_0(\delta ,\varepsilon )\ge 4\) for which all even \(N\ge N_0\) satisfy \(G_6+G_7-G_{12}+(c_0-C_9-10191/100000-\delta )\mathcal X_N\le 4D_{1,19/10}(N)\), at the fixed exponents of this chapter.

For every \(\delta {\gt}0\) and every fixed \(0{\lt}\varepsilon {\lt}2/15\), there is \(N_0\ge 4\) such that all even \(N\ge N_0\) satisfy \(G_9\le (C_9+\delta )\mathcal X_N\), with \(G_9=S_5(N^{4/53},N^{1/3})\). The threshold follows \(\delta ,\varepsilon \). The low part of \(C_9\) retains the additional factor \(1/(1-u)\), so its reduced kernel has denominator \(u(1-u)^2\).

For every fixed \(\varepsilon {\gt}0\), there is \(N_0\in \mathbb N\) such that, for all \(N\ge N_0\) and every real \(a\), \(\sum _{n\in \mathcal A}w_a(n)\le D_{1,19/10}(N)\). The threshold precedes \(a\); no evenness hypothesis is needed.

\(\forall \kappa {\lt}515093/800000000\ \exists K\ge 4\ \forall N\ge K,\ 2\mid N\Rightarrow \kappa \mathcal X_N\le D_{1,19/10}(N)\).

For every \(N\in \mathbb N\), \(D_{1,19/10}(N){\gt}0\) if and only if there exist \(p,r,q\in \mathbb N\) with \(p\le N\), \(p,q\) prime, \(r=1\) or \(r\) prime, \(N=p+rq\), and \(r^{10}\le q^9\).

For every \(\delta {\gt}0\) and every fixed \(0{\lt}\varepsilon \le 2/15\), there exists \(N_0(\delta ,\varepsilon )\ge 4\) such that all even \(N\ge N_0\) satisfy \(G_{12}\le (C_{12}+\delta )\mathcal X_N\). Both exception payments are included. The original domain is \(z\le r\le q\le s\le B\le t\le C\), and \(C_{12}\) uses the strict-low, closed-high Buchstab-weight split defined in the text.

\(C_{12}\le 66821/100000\).

\(I_{11}(h)=\frac{36}{5}\int _a^{1/10}\frac{K_{11}(u)}{1-u}\, du+8\int _{1/10}^bK_{11}(u)\, du\).

There exists \(K\ge 4\) such that every even \(N\ge K\) admits \(p,r,q\in \mathbb N\) with \(p\le N\), \(p,q\) prime, \(r=1\) or \(r\) prime, \(N=p+rq\), and \(r\le q^{(19/10)-1}\). This endpoint is supplied by the independent old-margin existence route.

For every \(\delta {\gt}0\) and each fixed \(0{\lt}\varepsilon {\lt}1\), some \(N_0\ge 4\) gives \(G_{10}^{\rm corr}\le [8(1-\varepsilon )I_{10}+\delta ]\mathcal X_N\) for every even \(N\ge N_0\), at \((b,c)=(4/33,3/11)\). The corrected fibre excludes prime divisors of \(Nrs\) and retains both labels; \(\mathcal X_N=\mathfrak S_{\rm Liu}(N)N/\log ^2N\).

For every \(\delta {\gt}0\) and every fixed \(0{\lt}\varepsilon \le 1\), there exists \(N_0(\delta ,\varepsilon )\ge 4\) such that all even \(N\ge N_0\) satisfy \(G_{11}\le (10191/100000+\delta )\mathcal X_N\) on the original domain \(z\le r\le q\le s\le t\le B\). No factor \(1-\varepsilon \) is asserted: this bound uses the full upper-endpoint rough mother.

\((561522/10^6)I_{11}(h)\le 10191/100000\).

For every \(\delta {\gt}0\) and each fixed \(0{\lt}\varepsilon {\lt}2/15\), some \(N_0\ge 4\) works for all even \(N\ge N_0\) and every \(1/18{\lt}a{\lt}4/33\): \(\mathcal T_{11}-[8(1-\varepsilon )I_{10}+\delta ]\mathcal X_N\le 4D_{1,19/10}(N)\). Here \(\mathcal T_{11}=3G_1+G_2-4G_3-G_4-G_5+G_6+G_7-2G_8-G_9-G_{11}-G_{12}\), with \(b=4/33,c=3/11\). Both the corrected tenth cost and \(1334N^{1-a}\) are paid.

For \(N,n,r,q\in \mathbb N\) and \(\varepsilon {\gt}0\), assume \(r{\gt}0\), \(n=rq{\gt}\varepsilon N\), \(r{\lt}\lceil N^{9/19-\varepsilon }\rceil \), and \((N^{9/19-\varepsilon })^{19}\le (\varepsilon N)^9\). Then \(r^{10}\le q^9\). The cutoff is equivalent to \(r{\lt}N^{9/19-\varepsilon }\) because \(r\) is integral.

For every \(\delta {\gt}0\), there is \(\varepsilon _0\in (0,2/15]\) such that, for every fixed \(0{\lt}\varepsilon {\lt}\varepsilon _0\), there exists \(N_0(\delta ,\varepsilon )\ge 4\) for which all even \(N\ge N_0\) satisfy \((515093/200000000-\delta )\mathcal X_N\le 4D_{1,19/10}(N)\). The threshold is not asserted uniformly over all small positive epsilon.

For \(N\ge 4\), \(N^{4/53}\ge 4\), \(1{\lt}\rho \le 5/4\), and \(0\le \eta {\lt}1/4\), every used grid cell and every short prime \(p\) in it satisfy \(4\log N/\log Q_{\mathrm{mix}}\le h(\log p/\log N)+32\eta \). The finite comparison has arbitrary real \(\varepsilon \), constrained only through membership in the used-cell set; \(\eta \) is the level loss, not a small-epsilon-window parameter.

For \(m_{\mathrm{old}}:=4J_{\mathrm{elem}}+c_0-C_9-10385101/10^8-C_{12}\), where \(J_{\mathrm{elem}}\) is the five fixed sum-coordinate integrals in the text (including the half-square multiplicity), \(126891/200000000\le m_{\mathrm{old}}\). This is an unconditional scalar certificate with no epsilon parameter, not the definition of the margin and not the author-route ledger.

For every \(\delta {\gt}0\), there exists \(\varepsilon _0\in (0,2/15]\) such that, for every fixed \(0{\lt}\varepsilon {\lt}\varepsilon _0\), there is \(N_0(\delta ,\varepsilon )\ge 4\) for which all even \(N\ge N_0\) satisfy \((C_{67}-\delta )\mathcal X_N\le G_6+G_7\), at \((a,b,c)=(4/53,4/33,3/11)\). The small epsilon window absorbs the source-mass loss before the natural threshold is chosen.

\(54233/10000\le C_{67}\).

\(\exists K\ge 4\ \forall N\ge K,\ 2\mid N\Rightarrow 0.0004\mathcal X_N{\lt}D_{1,19/10}(N)\).

For every \(\delta {\gt}0\) and every fixed \(0{\lt}\varepsilon {\lt}1\), there exists \(N_0\ge 4\) such that all even \(N\ge N_0\) satisfy \(((1-\varepsilon )L-\delta )\mathcal X_N\le 3G_1+G_2\), at \((a,b)=(4/53,4/33)\). The threshold may depend on both \(\delta \) and \(\varepsilon \); epsilon need not shrink with delta.

For every real \(\kappa {\lt}515093/800000000\), there exists \(K\in \mathbb N\), \(4\le K\), such that every even \(N\ge K\) satisfies \(\kappa \mathcal X_N\le D_{1,19/10}(N)\). The threshold follows the chosen coefficient; the ceiling itself is excluded.

\(\exists K\in \mathbb N\), \(4\le K\), \(\forall N\ge K\), \(2\mid N\Rightarrow (1/2500)\mathcal X_N{\lt}D_{1,19/10}(N)\). Here \(D\) counts distinct primes \(p\) and \(\mathcal X_N=\mathfrak S_{\rm Liu}(N)N/\log ^2N\), with no extra factor two.

\(\exists K\in \mathbb N\), \(4\le K\), such that for all even \(N\ge K\) there exist \(p,r,q\in \mathbb N\) with \(p\le N\), \(p,q\) prime, \(r=1\) or \(r\) prime, \(N=p+rq\) and \(r^{10}\le q^9\).

\(\exists K\in \mathbb N\), \(4\le K\), such that for all even \(N\ge K\) there exist \(p,r,q\in \mathbb N\) with \(p\le N\), \(p,q\) prime, \(r=1\) or \(r\) prime, \(N=p+rq\) and \((r:\mathbb R)\le (q:\mathbb R)^{(19/10)-1}\).

For \(N,p\in \mathbb N\), the representation predicate is \(p\) prime and \(\exists r,q\in \mathbb N\): \((r=1\text{ or }r\text{ prime})\), \(q\) prime, \(N=p+rq\), and \(r^{10}\le q^9\).

For every \(\delta {\gt}0\), there exists \(\varepsilon _0\in (0,2/15]\) such that, for every fixed \(0{\lt}\varepsilon {\lt}\varepsilon _0\), there is \(N_0(\delta ,\varepsilon )\ge 4\) for which all even \(N\ge N_0\) satisfy \(G_6+G_7-G_9-G_{11}-G_{12}+(124341093/200000000-\delta )\mathcal X_N\le 4D_{1,19/10}(N)\), at the fixed exponents of this chapter.

For every \(\rho {\gt}0\), some \(N_0\ge 2\) works for all even \(N\ge N_0\) and every \(0{\lt}\varepsilon {\lt}1\): \((f(6)-\rho )V_N\le S_N.\mathrm{mainSum}(\lambda ^-)\), where \(S_N\) is the actual \(S_1\) bounding sieve at \(z=N^{4/53}\), \(V_N\) its sieve product and \(\lambda ^-\) its lower Rosser weight at \(D=\lceil N^{4/53}\rceil ^6\). The density threshold precedes epsilon.

For every \(\rho {\gt}0\), some \(z_0\ge 2\) works uniformly in \(N\), its evenness proof, and real \(\varepsilon ,z,\Delta ,s\): if \(z\ge z_0\), \(\Delta {\gt}0\) and \(s=\log \Delta /\log z\in [3/2,6]\), then \(\sum _{d\mid P_N(z)}\lambda _d^+/\varphi (d)\le (F_S(s)+\rho )V_N(z)\). The natural level is \(\lfloor \Delta \rfloor +1\) and the local-product witness is supplied internally.

For \(N\ge 2\), \(\varepsilon {\gt}0\), \(\beta {\gt}1/18\), any real second cutoff \(C\), and \(Z\ge 1\), the finite labelled counts with first cutoff \(B=N^\beta \) satisfy \(\Pi _{10}\le \mathcal B_{10}(Z)+400\lfloor Z\rfloor \). Here the output \(N-rsq\) is sifted excluding prime divisors of \(N\), while the pair labels and strict prime-cofactor window are retained. Neither eventuality nor evenness is required.

Fix \(0{\lt}\varepsilon {\lt}2/15\). There is \(N_0\in \mathbb N\) such that every even \(N\ge N_0\) and every \(1/21{\lt}k{\lt}\sigma \le 1/3\), with \(z=N^k,y=N^\sigma ,T=N^{9/19-\varepsilon }\), satisfy \(2S_1(z)-2S_2(T)-S_3(z,y)-2S_4(y)-S_5(z,y)+S_6(z,y)-446N^{1-k}\le 2D_{1,19/10}(N)\). This threshold is uniform in \(k,\sigma \); there is no additional positive-loss parameter.

Fix \(0{\lt}\varepsilon {\lt}2/15\). One threshold works for every \(1/18{\lt}a{\lt}b{\lt}(1-3b)/3{\lt}c{\lt}1/3\). With the corresponding cutoffs, every even \(N\) beyond it satisfies \(3G_1+G_2-4G_3-G_4-G_5+G_6+G_7-2G_8-G_9-G_{10}^{\mathrm{corr}}-G_{11}-G_{12}-1334N^{1-a}\le 4D_{1,19/10}(N)\). In the analytic application \((a,b,c)=(4/53,4/33,3/11)\).

For every \(K{\gt}1\) and \(\rho {\gt}0\), some \(z_0\) works for every bounding sieve \(S\) and real \(z,\Delta ,s\): if \(z\ge \max (2,z_0)\), \(\Delta {\gt}0\), \(S\) has the dimension-one local-product bound with \(K\), all its sifting primes are at most \(z\), and \(s=\log \Delta /\log z\in [3/2,6]\), then \(S.\mathrm{mainSum}(\lambda ^+)\le (F_S(s)+\rho )V_S\). Here \(F_S\) denotes the Suzuki factor, \(V_S\) the sieve product, and \(\lambda ^+\) has natural level \(\lfloor \Delta \rfloor +1\).