--- title: Zsiflaw–Legeis Theorem url: https://www.emergentmind.com/topics/zsiflaw-legeis-theorem type: topic --- # Zsiflaw–Legeis Theorem The term **Zsiflaw–Legeis Theorem** is used in the supplied literature in two distinct ways. In an earlier analytic-number-theoretic usage, it denotes a law of asymptotic equality between positive and negative signum-areas of Hardy’s \(Z\)-function over specially constructed disconnected sets [1312.4767]. In a later and now structurally different usage, it denotes a Siegel–Walfisz-type distribution theorem for digit reversals of primes in arithmetic progressions, first in fixed digit-length windows and then in a refined counting-by-size form [2507.08714]. The shared name is itself deliberate: in the prime-reversal literature, “Zsiflaw–Legeis” is “Siegel–Walfisz” reversed [2507.08714]. ## 1. Terminology and competing meanings The nomenclature is context-dependent. The two principal meanings appearing in the record are summarized below. | Usage | Core object | Source | |---|---|---| | Signum-area law | Asymptotic equality of positive and negative areas under \(Z(t)\) on \(G_1(x)\cup G_2(x)\) | [1312.4767] | | Reversed-prime distribution law | Equidistribution of digit reversals of primes in arithmetic progressions | [2507.08714] | In the Hardy \(Z\)-function setting, the name refers to the paper’s “law of asymptotic equality of signum-areas of \(Z(t)\),” namely the asymptotic balance of areas above and below the \(t\)-axis over the disconnected sets \(G_1(x)\cup G_2(x)\) [1312.4767]. In the digital-reversal setting, the name is explicitly motivated by analogy with Siegel–Walfisz: the theorem gives a quantitative distribution law for reversed primes in residue classes, with fixed-window and absolute-counting variants [2507.08714]. This suggests a genuine bifurcation of terminology rather than a single universally accepted theorem. In contemporary work on reversed primes, the phrase “Zsiflaw–Legeis” is used for arithmetic progression equidistribution of reversed primes, including a refined version adapted to additive problems [2605.21876]. ## 2. The signum-area Zsiflaw–Legeis theorem for Hardy’s \(Z\)-function In the 2013 usage, the ambient object is Hardy’s function \[ Z(t)=e^{i\theta(t)}\zeta\!\left(\tfrac12+it\right), \] where \[ \theta(t)=\arg\Gamma\!\left(\tfrac14+\tfrac{it}{2}\right)-\tfrac{t}{2}\log\pi. \] The paper works with Gram points \(t_\nu\) defined by \(\theta(t_\nu)=\pi\nu\), and shifted Gram points \(t_\nu(\tau)\) defined by \[ \theta\big(t_\nu(\tau)\big)=\pi\nu+\tau,\qquad t_\nu(0)=t_\nu. \] Using these, it introduces disconnected sets \[ G_{1}(x)=\bigcup_{T\le t_{2\nu}\le T+H}\big(t_{2\nu}(-x),\,t_{2\nu}(x)\big),\qquad G_{2}(y)=\bigcup_{T\le t_{2\nu+1}\le T+H}\big(t_{2\nu+1}(-y),\,t_{2\nu+1}(y)\big), \] in the short-interval regime \[ H=T^{1/6+2\varepsilon},\qquad \varepsilon>0\ \text{fixed and small},\qquad x,y\in(0,\pi]. \] The associated positive and negative signum-areas over a measurable set \(I\) are \[ A^{+}(I)=\int_{I\cap\{Z(t)>0\}} Z(t)\,dt,\qquad A^{-}(I)=\int_{I\cap\{Z(t)<0\}} |Z(t)|\,dt. \] The theorem itself states that, for \(T\to\infty\), \(H=T^{1/6+2\varepsilon}\), and uniformly for \(x\in(0,\pi]\), \[ A^{+}\big(G_{1}(x)\cup G_{2}(x)\big)=A^{-}\big(G_{1}(x)\cup G_{2}(x)\big)+O(xH). \] Equivalently, \[ \int_{G^{+}_{1}(x)\cup G^{+}_{2}(x)} Z(t)\,dt = -\int_{G^{-}_{1}(x)\cup G^{-}_{2}(x)} Z(t)\,dt +O(xH), \] and in geometric notation \[ m\{D^{+}(x)\}\sim m\{D^{-}(x)\},\qquad T\to\infty. \] The content is a balancing law: over the carefully tailored disconnected set \(G_1(x)\cup G_2(x)\), the positive and negative excursions of \(Z(t)\) contribute asymptotically equal area [1312.4767]. The theorem is built on mean-value asymptotics over the disconnected sets: \[ \int_{G_{1}(x)} Z(t)\,dt=\frac{2}{\pi}H\sin x+O\!\big(xT^{1/6+\varepsilon}\big),\qquad \int_{G_{2}(y)} Z(t)\,dt=\frac{2}{\pi}H\sin y+O\!\big(yT^{1/6+\varepsilon}\big), \] uniformly for \(x,y\in(0,\pi]\). These formulas decompose a short Hardy–Littlewood integral into contributions over \(G_1\) and \(G_2\), and for \(x=y=\pi/2\) recover \[ \int_T^{T+H} Z(t)\,dt=O(H). \] The signum-area theorem is therefore not an isolated geometric statement; it is the geometric reformulation of a fine mean-value theory for \(Z(t)\) on disconnected sets [1312.4767]. ## 3. The fixed-window Zsiflaw–Legeis theorem for reversals of primes In the later literature, the theorem concerns the digital reverse of primes. For a fixed base \(g\ge 2\), every \(n\ge 0\) has a base-\(g\) expansion \[ n=\sum_{i\ge 0}\varepsilon_i(n)g^i, \] with digit length \[ \operatorname{len}(n):=\min\{\ell\ge 0:\varepsilon_i(n)=0\ \text{for all } i\ge \ell\},\qquad \operatorname{len}(0)=0. \] The absolute digital reverse is \[ \operatorname{rev}(n):=\sum_{0\le i<\operatorname{len}(n)} \varepsilon_i(n)\,g^{\operatorname{len}(n)-i-1}, \] and the relative reverse in a fixed digit window of length \(L\) is \[ \operatorname{rev}_L(n):=\sum_{0\le i0\) and the implied constants depend only on \(g\) and are effectively computable [2507.08714]. This is the reversed-digit analogue of a Siegel–Walfisz theorem. The same paper also proves a reversed-digit analogue of Dirichlet’s theorem, called “Telhcirid’s theorem on arithmetic progressions”: for \(g\ge 2\), \(q\ge 1\), \((a,q,g^2-1)=1\), and \(g\nmid(a,q)\), there are infinitely many primes \(p\) such that \(\operatorname{rev}(p)\equiv a\pmod q\) [2507.08714]. A significant aspect of the 2025 result is the removal of earlier base-size restrictions. Previous work had required \(g\ge 31699\), later improved to \(g\ge 26000\), whereas the arbitrary-base theorem holds for every integer base \(g\ge 2\) [2507.08714]. ## 4. Weakly digital functions, proof architecture, and absolute-counting forms The arbitrary-base theorem is proved by extending the Martin–Mauduit–Rivat framework from digital functions to **weakly digital functions**, which may depend on both digit and position. For a seed \(\alpha\in\mathfrak A_g\), the basic phase is \[ f_{\lambda,\alpha}(n):=\sum_{0\le i<\lambda}\alpha_i(\varepsilon_i(n)). \] Reversal is modeled by taking, for fixed \(L\) and \(\alpha\in\mathbb R\), \[ \alpha_{L,i}(n):=\alpha\,n\,g^{L-i-1}, \] so that \[ f_{L,\alpha_L}(n)=\alpha\,\operatorname{rev}_L(n),\qquad e(f_{L,\alpha_L}(n))=e(\alpha\,\operatorname{rev}_L(n)). \] The key prime-exponential-sum theorem has the shape \[ \sum_{n\le x}\Lambda(n)e(f_{L,\alpha}(n)) \ll x\,g^{-\kappa}(\log x)^4, \] where \[ \kappa=\tfrac{1}{10}\sigma_\xi(\alpha),\qquad \xi=\left\lfloor \tfrac14\frac{\log x}{\log g}\right\rfloor. \] Its proof combines product formulas for normalized sums \(F_\lambda^{[j]}(\beta)\), pointwise \(L^\infty\) bounds, discrete \(L^1\) bounds, a hybrid large-sieve type estimate, and Type I and Type II bilinear bounds obtained through Vaughan’s identity [2507.08714]. The distribution theorem for \(\operatorname{rev}_L\) is then derived by additive characters. Major arcs correspond to the condition that the congruence collapses to an essential modulus \((q,g^L(g^2-1))\), while minor arcs are controlled through the weakly digital prime-exponential-sum estimate. This yields Siegel–Walfisz-type formulas for \[ \psi_L(x,a,q),\qquad \theta_L(x,a,q),\qquad \pi_L(x,a,q), \] uniformly for \[ q\le \exp\!\left(c\,\frac{\log x}{\log\log x}\right), \] with error \[ O\!\left(x\exp\!\left(-c\,\frac{\log x}{\log(q+1)}\right)\right). \] The same paper also gives a “pure” absolute version, counting primes up to \(x\) by their actual reversal rather than inside a fixed window. If \[ L=\left\lfloor \frac{\log x}{\log g}\right\rfloor+1, \] then \[ \overleftarrow{\pi}(x,a,q) = \frac{(q,(g^2-1)g^L)}{q}\,\overleftarrow{\pi}_{\#}(x,a,q) + O\!\left(x\exp\!\left(-c\,\frac{\log x}{\log(q+1)}\right)\right), \] uniformly for \[ q\le \exp\!\left(c\,\frac{\log x}{\log\log x}\right). \] Thus the fixed-window theorem and the absolute-counting theorem are two layers of the same analytic structure: the first isolates stable digit-length blocks, while the second sums those blocks into a global counting statement [2507.08714]. ## 5. The refined Zsiflaw–Legeis theorem without fixed digit length The 2026 refinement reformulates the reversed-prime theorem in a form directly suited to additive problems. For a fixed base \(b\ge 2\), write \[ \operatorname{Rev}_b(n)=\sum_{i=0}^k d_i\,b^{k-i} \] when \[ n=\sum_{i=0}^k d_i b^i,\qquad d_k\ne 0. \] If \(p\) is prime, its reversal is \[ \overleftarrow p:=\operatorname{Rev}_b(p). \] The weighted progression count is \[ \overleftarrow{\vartheta}(X;q,a):= \sum_{\substack{\overleftarrow p\le X\\ \overleftarrow p\equiv a\ (\mathrm{mod}\ q)}} \log p. \] The earlier fixed-\(L\) theorem in this literature has the asymptotic shape \[ \overleftarrow{\vartheta}_L(a,q) = \frac{\varphi(b)}{b}\cdot \frac{(q,b^3-b)}{\varphi((q,b^3-b))}\cdot \frac{\rho_b(a,q)}{q}\cdot b^L + O_b\!\big(b^L e^{-c\sqrt L}\big), \] uniformly for \[ q\le e^{c\sqrt L}, \] where \[ \rho_b(a,q)= \begin{cases} 1 & \text{if } (a,q,b^3-b)=1,\\ 0 & \text{if } (a,q,b^3-b)>1. \end{cases} \] A modified effective variant restricts to reversals coprime to \(b^3-b\) and yields the same main term and error, again for \(q\le e^{c\sqrt L}\) [2605.21876]. The paper’s main contribution is the **refined Zsiflaw–Legeis theorem without fixing \(L\)**. Let \[ \mathfrak B(x):=\{n\le x:(\operatorname{Rev}_b(n),b)=1\}. \] Then, for any \(b\ge 2\), \(q\ge 1\), \((a,q)=1\), and \(A>0\), \[ \overleftarrow{\vartheta}^{*}(x;a,q) := \sum_{\substack{\overleftarrow p\le x\\ \overleftarrow p\equiv a\ (\mathrm{mod}\ q)\\ (\overleftarrow p,b^3-b)=1}} \log p = \frac{(q,b^3-b)}{\varphi((q,b^3-b))}\cdot \frac{\rho_b(a,q)}{q}\cdot |\mathfrak B(x)| + O_{b,A}\!\left(\frac{x}{(\log x)^A}\right), \] uniformly for \[ q\le e^{c\sqrt L},\qquad b^{L-1}0\) is ineffective because the argument uses Siegel–Walfisz for primes [2605.21876]. Three structural features are encoded directly in the formula. First, the leading-digit constraint is absorbed by \(|\mathfrak B(x)|\). Second, the local obstruction modulo \(b^3-b\) appears through \[ \frac{(q,b^3-b)}{\varphi((q,b^3-b))}\cdot \rho_b(a,q). \] Third, the theorem is specifically designed to remove the fixed-digit-length restriction that obstructs circle-method applications [2605.21876]. ## 6. Additive consequences, effectivity, and common misconceptions The refined theorem feeds directly into a Hardy–Littlewood circle-method analysis of mixed prime/reversed-prime representation problems. The paper proves four headline consequences: every large odd integer is the sum of a prime and two reversed primes; every large odd integer is the sum of two primes and a reversed prime; almost all even integers are the sum of a prime and a reversed prime; and all large integers are the sum of a reversed prime and a square-free number [2605.21876]. More precisely, for the ternary problems one obtains asymptotics of the form \[ \mathcal R_{1,2}(N)=\mathfrak S_3(N)\,\mathcal S_{1,2}(N)+O_{b,A}\!\left(\frac{N^2}{(\log N)^A}\right), \] and \[ \mathcal R_{2,1}(N)=\mathfrak S_3(N)\,\mathcal S_{2,1}(N)+O_{b,A}\!\left(\frac{N^2}{(\log N)^A}\right). \] For the binary problem, all but \[ O_{b,A}\!\left(\frac{x}{(\log x)^A}\right) \] even integers \(N\le x\) are representable as \[ N=p+\overleftarrow p, \] with \((\overleftarrow p,b^3-b)=1\). For the square-free complement problem, \[ \mathcal R_{\square}(N) = \frac{|\mathfrak B(N)|}{\zeta(2)}\cdot \mathfrak S_{\square}(N) + O_{b,A}\!\left(\frac{N}{(\log N)^A}\right), \] which implies representations \[ N=\overleftarrow p+\eta,\qquad \mu^2(\eta)=1, \] for all sufficiently large \(N\) [2605.21876]. Methodologically, the major arcs use the classical prime sum \[ S(\alpha)=\sum_{p\le N} e(p\alpha)\log p \] together with the reversed-prime sum \[ \overleftarrow S_b(\alpha)= \sum_{\substack{\overleftarrow p\le N\\ (\overleftarrow p,b^3-b)=1}} e(\overleftarrow p\alpha)\log p, \] and the refined Zsiflaw–Legeis theorem supplies the major-arc asymptotics for \(\overleftarrow S_b(\alpha)\). Minor arcs use Vinogradov-type bounds for \(S(\alpha)\) and Parseval-based \(L^2\) control for \(\overleftarrow S_b(\alpha)\) [2605.21876]. Two misconceptions are especially common. The first is that “Zsiflaw–Legeis theorem” names a single universally fixed result. The literature supplied here does not support that reading: one usage concerns signum-areas of Hardy’s \(Z\)-function, while another concerns arithmetic progression equidistribution of reversed primes [1312.4767]. The second is that all versions have the same analytic status. The arbitrary-base fixed-window theorem is stated with effective constants depending only on \(g\) [2507.08714], whereas the refined counting-by-\(x\) theorem is ineffective because it relies on Siegel–Walfisz for primes [2605.21876]. Taken together, these results show that the name now covers two unrelated but structurally parallel themes: a geometric balancing law for oscillatory values of \(Z(t)\), and a reversed-digit analogue of classical prime distribution theorems. The latter has become the central meaning in recent additive work on reversed primes [2605.21876]. Source: https://www.emergentmind.com/topics/zsiflaw-legeis-theorem