--- title: Quasi-Riemann Hypothesis url: https://www.emergentmind.com/topics/quasi-riemann-hypothesis type: topic --- # Quasi-Riemann Hypothesis The **Quasi-Riemann Hypothesis** is not a single standard mathematical statement. In the literature represented by the cited arXiv papers, the expression denotes several distinct constructions: a weakened zero-free assertion such as $\sup_\rho \Re(\rho)<1$, an RH-type conjecture for zeta-like functions generated by continued-fraction permutations, or an alternative formulation of the classical Riemann Hypothesis through generalized functions, probability, harmonic analysis, or zero-free regions. These usages must be distinguished from the classical Riemann Hypothesis (RH), which asserts that every nontrivial zero $\rho$ of $\zeta(s)$ satisfies $\Re(\rho)=\frac12$. ## 1. Terminological scope and classical reference point The Riemann zeta function is initially defined for $\Re(s)>1$ by \[ \zeta(s)=\sum_{n=1}^{\infty}n^{-s} =\prod_p(1-p^{-s})^{-1}, \] and extends meromorphically to $\mathbb C$, with a simple pole at $s=1$. Its nontrivial zeros lie in the critical strip \[ 0<\Re(s)<1, \] and occur symmetrically under \[ \rho\longmapsto 1-\rho,\qquad \rho\longmapsto\overline{\rho}. \] The completed function \[ \xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma\!\left(\frac{s}{2}\right)\zeta(s) \] is entire, satisfies \[ \xi(s)=\xi(1-s), \] and has precisely the nontrivial zeros of $\zeta(s)$, with multiplicity. The classical Riemann Hypothesis is \[ \mathrm{RH}:\qquad \zeta(\rho)=0,\quad 0<\Re(\rho)<1 \Longrightarrow \Re(\rho)=\frac12. \] The phrase “Quasi-Riemann Hypothesis” has at least three principal meanings in the cited research: 1. **A weakened zero-location assertion**, especially $\sup_\rho\Re(\rho)<1$. 2. **A generalized RH for modified zeta-like functions**, such as continued-fraction permutation zeta functions. 3. **An equivalent reformulation of RH**, using generalized functions, characteristic functions, positivity inequalities, or harmonic boundary-value problems. The 2017 paper “Average Goldbach and the Quasi-Riemann Hypothesis” explicitly defines QRH as \[ \boxed{\sup_{\rho}\Re(\rho)<1}, \] where $\rho$ ranges over the nontrivial zeros of $\zeta(s)$ [1711.06442]. The 2022 paper “On the complex zeros of the Riemann Zeta-function” uses the same formulation and studies the conditional implication \[ \mathrm{QRH}\Longrightarrow\mathrm{RH} \] [2210.03121]. By contrast, the 2026 paper on uniform zero-free half-planes uses QRH in a family-wide sense: the existence of a fixed $B<1$ such that a collection of Hecke or Dirichlet $L$-functions is zero-free in $\Re(s)>B$ [2610.12234]. Thus, QRH is a polysemous label rather than a universally accepted named conjecture. ## 2. The weakened zero-free formulation In the weakest zeta-function usage, QRH asserts that all nontrivial zeros lie uniformly to the left of the line $\Re(s)=1$: \[ \boxed{\sup_\rho\Re(\rho)<1.} \] Equivalently, there exists some $a<1$ such that every nontrivial zero satisfies $\Re(\rho)\le a$. This condition is strictly weaker, as a statement of zero location, than RH. It does not require $\Re(\rho)=\frac12$. The 2017 Goldbach formulation considers the weighted Goldbach representation function \[ G(n)=\sum_{k_1+k_2=n}\Lambda(k_1)\Lambda(k_2), \] its summatory function \[ S(x)=\sum_{n\le x}G(n), \] and the generating function \[ F(z)=\sum_{n=1}^{\infty}\Lambda(n)z^n. \] The identity \[ F(z)^2=(1-z)\sum_{n=1}^{\infty}S(n)z^n \] connects an average Goldbach estimate with the prime-counting function. The principal hypothesis is \[ S(x)=\frac{x^2}{2}+O(x^{2-\delta}),\qquad \delta>0. \] The paper proves that this implies the existence of $\delta'>0$ such that every nontrivial zero satisfies \[ \Re(\rho)<1-\delta'. \] The proof as presented yields \[ \delta'=\frac{\delta}{6}, \] while a more careful kernel is said to improve this to $\delta'=\frac{\delta}{3}$ [1711.06442]. The argument proceeds through the approximation \[ F(z)^2=\frac{1}{(1-z)^2}+\text{controlled error}, \] followed by a square-root estimate \[ F(z)=\frac{1}{1-z}+\text{controlled error}. \] A Cauchy integral with kernel \[ K(z)=z^{-N-1}\frac{1-z^N}{1-z} \] then gives \[ \psi(N)=N+O\!\left(N^{1-\delta/6}(\log N)^{1/2}\right). \] The explicit formula for $\psi(x)$ associates a zero $\rho=\beta+i\gamma$ with an oscillatory contribution of size approximately $x^\beta$. A power-saving estimate for $\psi(x)-x$ therefore excludes zeros with sufficiently large real part. The logical content is \[ S(x)=\frac{x^2}{2}+O(x^{2-\delta}) \Longrightarrow \Re(\rho)<1-\delta'. \] The paper also cites results giving an implication from RH to a sufficiently strong average Goldbach estimate. However, the converse proved by the generating-function argument yields a zero-free half-plane, not directly $\Re(\rho)=\frac12$. Consequently, the literal condition $\Re(\rho)<1$ should not be identified with RH. The 2022 paper attempts to strengthen this weak assertion by proving QRH $\Rightarrow$ RH. It defines \[ b=\sup_{\gamma>10^3}\left(\beta-\frac12\right) \] for zeros $\rho=\beta+i\gamma$ and constructs finite Möbius polynomials \[ M_V(s)=\sum_{n\le V}\frac{\mu(n)}{n^s}. \] A zero $s_V$ of $M_V$ near $1$ is used to form \[ F_V(s)=\zeta(s)\,M_V(s+s_V-1), \] which cancels the pole of $\zeta(s)$ at $1$ while retaining the zeta zeros. Symmetrization around a selected large zero produces \[ G_{U,V}(s)=U^{s-s_0}\frac{F_V(s+iv)+F_V(s-iv)}2. \] A high-order Taylor expansion, together with estimates for the derivatives of $G_{U,V}$, is then claimed to contradict the existence of a zero with $\beta>\frac12$ under the assumption $\sup_\rho\beta<1$ [2210.03121]. The result is conditional: it does not establish QRH itself. ## 3. Uniform QRH for families of $L$-functions A broader family-theoretic meaning of QRH appears in “Slightly improved zero-free half-planes for the quasi-Riemann hypothesis.” There the objective is to establish a uniform boundary $B<1$ such that \[ L_F(s,\eta)\neq0\qquad(\Re s>B) \] for every primitive finite-order Hecke character $\eta$ over \[ F=\mathbb Q(\sqrt{-3}), \] with arbitrary conductor. The same boundary is transferred to Dirichlet $L$-functions and hence to $\zeta(s)$ [2610.12234]. The family-wide extremal quantity is \[ \beta_*=\sup\left( \left\{\frac12\right\} \cup \left\{\Re\rho: L_F(\rho,\eta)=0,\ \eta\ \text{primitive and finite-order} \right\} \right). \] A bound $\beta_*\le B$ means that every member of the family is zero-free in the open half-plane $\Re(s)>B$; no assertion is made about the boundary line $\Re(s)=B$. The paper records three successive bounds: \[ \beta_*\le\frac78, \] then \[ \beta_*\le \frac{34999}{40000}=0.874975, \] and finally \[ \boxed{ \beta_*\le B_{\mathrm{new}} = \frac{1507-2\sqrt{921}}{1653} = 0.874957069799\ldots }. \] The underlying method uses a cubic metaplectic theta average, Poisson summation, reciprocal Hecke $L$-functions, selected prime modifications, character-polynomial detectors, moment estimates, and a Mellin-continuation contradiction. The scale parameters are \[ b=l_y-l_x,\qquad l_x=\frac{1-b-\ell}{2},\qquad l_y=\frac{1+b-\ell}{2}, \] with selected prime lengths satisfying \[ \prod_iP_i=Z^\ell. \] The comparison boundary is \[ B(\ell)=\frac{11}{12}-\frac{\ell}{4}. \] The rational improvement fixes \[ b_{\mathrm r}=\frac18,\qquad \ell_{\mathrm r}=\frac16+\frac1{10000}, \] giving \[ B_{\mathrm r}=\frac78-\frac1{40000}. \] The optimized improvement varies both $b$ and $\ell$ and produces the radical $\sqrt{921}$ through a critical exponent equation. The paper states that the rational and algebraic parameter calculations were formalized in Lean, while the analytic estimates on which the argument depends are imported or abstracted inputs [2610.12234]. This family-theoretic QRH is a uniform zero-free-region problem. It is not the assertion that all zeros lie on a critical line. In particular, \[ \beta_*<1 \] is compatible with zeros having real parts strictly greater than $\frac12$. ## 4. Generalized RH-type functions from continued fractions A different usage arises from “Yet Another Riemann Hypothesis,” which proposes a continued-fraction permutation generalization of the zeta integral rather than a weakened assertion about the zeros of $\zeta(s)$ [1101.0311]. The classical Gauss-map representation is \[ \zeta(s)=\frac{s}{s-1}-s\int_0^1h(x)x^{s-1}\,dx, \] where \[ h(x)=\frac1x-\left\lfloor\frac1x\right\rfloor. \] If \[ x=[a_1,a_2,a_3,\ldots], \] then \[ h(x)=[a_2,a_3,\ldots], \] so $h$ is the continued-fraction shift map. The proposed replacement acts on the continued-fraction digits. For positive integers $p,q$, \[ S_{p,q}:[a_1,\ldots,a_p,\ldots,a_q,\ldots] \longmapsto [a_1,\ldots,a_q,\ldots,a_p,\ldots]. \] The sequence space is represented as \[ \mathbb M^\omega,\qquad \mathbb M=\mathbb N\cup\{\infty\}, \] with a quotient \[ \mathbb S=\mathbb M^\omega/\ker[\,] \] introduced to handle nonunique continued-fraction expansions. The associated zeta-like function is \[ \boxed{ \zeta_{p,q}(s)= \frac{s}{s-1} -s\int_0^1S_{p,q}(x)x^{s-1}\,dx. } \] The proposed quasi-Riemann-type assertion is \[ \boxed{ \zeta_{p,q}(s)=0 \Longrightarrow \Re(s)=\frac12 } \] for the relevant nontrivial zeros. The maps $S_{p,q}$ are stated to map the unit interval onto itself, to have countably many discontinuities, and to be piecewise Möbius: \[ S_{p,q}(x)=\frac{ax+b}{cx+d}, \qquad a,b,c,d\in\mathbb Z,\qquad ad-bc=\pm1. \] Their graphs have fractal or self-similar structure. The ordinary Gauss map is not itself one of the finite transpositions; it serves as the comparison map. The main computational approximation is midpoint quadrature: \[ \zeta_{N;p,q}(s) = \frac{s}{s-1} -\frac{s}{N}\sum_{n=0}^{N-1} S_{p,q}\!\left(\frac{2n+1}{2N}\right) \left(\frac{2n+1}{2N}\right)^{s-1}. \] The paper studies $|\zeta_{N;p,q}(s)|$ in regions with approximately \[ 13\le\Im(s)\le34, \] using values such as \[ N=151051. \] The finite sums exhibit artifact zeros. As $N$ increases, some features approach the critical line while other features fade or acquire weak visual intensity. For $S_{1,2}$, the dominant features approximately correspond to those of the Riemann zeta function, although the apparent zeros are visibly shifted. For $S_{1,3}$ and $S_{2,3}$, the correspondence is less clear because the discontinuity structure is finer than the numerical grid resolves. The paper estimates the first apparent zero of $\zeta_{1,2}$ near \[ s\approx\frac12+14.92\,i. \] Its location fluctuates strongly with $N$, and the real part is substantially less stable than the imaginary part. A possible oscillatory component with period approximately \[ \Delta N\approx300 \] and an approximately $1/f$-type power spectrum are reported. A calibration with $p=q$ gives \[ S_{p,p}(x)=x \] and \[ \lim_{N\to\infty}\eta_N(s)=\frac{2s}{s^2-1}. \] Although the limiting function is elementary and does not encode the Riemann zeros, finite approximations exhibit “shadow” features aligned with Riemann zeros. This occurs because the midpoint sums can be rewritten using truncated zeta sums. The experiment demonstrates that visual alignment in finite numerical approximations cannot establish exact zero correspondence. No Dirichlet-series representation, Euler product, completed function, functional equation, rigorous critical strip, analytic continuation theorem, or zero-location theorem is derived for $\zeta_{p,q}$. The critical line is imported by analogy with $\zeta(s)$ rather than forced by a proved functional equation. ## 5. Equivalent formulations of classical RH Several cited papers use “quasi” in a methodological rather than weakened sense: they replace RH by an equivalent condition without changing the underlying zero statement. ### 5.1 Deformation and two-variable Borel transforms For the Riemann $\Xi$-function, one normalization is \[ \Xi(z)=\frac18(z^2-1)\pi^{-z/2-1} \Gamma\!\left(\frac{z+1}{4}\right) \zeta\!\left(\frac{z+1}{2}\right) = \sum_{n\ge0}(-1)^nA_nz^{2n}, \qquad A_n>0. \] Defining \[ f^b(x)=\Xi(-i\sqrt{x})=\sum_{n\ge0}A_nx^n, \] RH is equivalent to all zeros of $f^b$ being negative real numbers. The two-variable $y$-Borel transform is \[ B_y(f^b)(x)= \sum_{n\ge0}A_nx^ny^{n(n+1)/2}. \] The paper proves that for $|y|<\rho_0$, with \[ \rho_0=0.2078\ldots, \] the zeros of $B_y(f^b)$ are represented by convergent power series with real coefficients. For real $01$, the paper proves that $\Xi_\sigma$ is quasi-infinitely divisible but not infinitely divisible: the associated Lévy measure is signed, although it satisfies the total-variation integrability condition [1504.03438]. ### 5.3 Positivity criteria and generalized Li sums For real $b\ne-\frac12$, generalized Li sums are \[ k_{n,b} = \sum_\rho \left[ 1- \left( 1-\frac{\rho+b}{\rho-b-1} \right)^n \right]. \] The generalized Bombieri–Lagarias criterion gives \[ \boxed{ \mathrm{RH} \iff k_{n,b}\ge0 \quad\text{for every }n\ge1 } \] for each admissible fixed $b$, with equivalent formulations using the entire parameter family. Under RH, the asymptotic is \[ k_{n,b} = \frac12|2b+1|\,n\log n + \frac12|2b+1| \left( \gamma-1-\log\frac{2\pi}{|2b+1|} \right)n +o(n). \] The parameter changes the positivity functional and its growth scale, but the zero-location assertion remains the ordinary RH [1403.4484]. ### 5.4 Colombeau–Beurling formulations The Nyman–Beurling and Baez–Duarte criteria characterize RH through the closure of the Beurling functions \[ \varphi_{1/k}(x)=\left\{\frac1{kx}\right\}, \qquad 0