Quasi-Riemann Hypothesis

• The Quasi-Riemann Hypothesis (QRH) refers to multiple distinct formulations, including weakened zero-free assertions, generalized Riemann Hypothesis (RH) for modified zeta-like functions, and equivalent reformulations of the classical RH.
• Applications and significance vary widely by formulation: For example, QRH as $\sup\_\rho\Re(\rho)<1$ serves as a guide for bounding nontrivial zeros' location of the Riemann zeta function, whereas in zeta-like functions it implies specific cases of zero locations but not strictly $\Re(s)=\frac{1}{2}$.
• Examples of QRH include weaker formulations such as\boxed{\sup\_{\rho}\Re(\rho)<1} for average Goldbach estimates in Studying number theory and prime quantity distribution.What does the QRH as LH implies?

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

$\rho$0

is entire, satisfies

$\rho$1

and has precisely the nontrivial zeros of $\rho$2, with multiplicity.

The classical Riemann Hypothesis is

$\rho$3

The phrase “Quasi-Riemann Hypothesis” has at least three principal meanings in the cited research:

1. A weakened zero-location assertion, especially $\rho$4.
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

$\rho$5

where $\rho$6 ranges over the nontrivial zeros of $\rho$7 (Bhowmik et al., 2017). The 2022 paper “On the complex zeros of the Riemann Zeta-function” uses the same formulation and studies the conditional implication

$\rho$8

(Puglisi, 2022). By contrast, the 2026 paper on uniform zero-free half-planes uses QRH in a family-wide sense: the existence of a fixed $\rho$9 such that a collection of Hecke or Dirichlet $\zeta(s)$0-functions is zero-free in $\zeta(s)$1 (Liu, 8 Oct 2026).

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 $\zeta(s)$2: $\zeta(s)$3 Equivalently, there exists some $\zeta(s)$4 such that every nontrivial zero satisfies $\zeta(s)$5. This condition is strictly weaker, as a statement of zero location, than RH. It does not require $\zeta(s)$6.

The 2017 Goldbach formulation considers the weighted Goldbach representation function

$\zeta(s)$7

its summatory function

$\zeta(s)$8

and the generating function

$\zeta(s)$9

The identity

$\Re(\rho)=\frac12$0

connects an average Goldbach estimate with the prime-counting function.

The principal hypothesis is

$\Re(\rho)=\frac12$1

The paper proves that this implies the existence of $\Re(\rho)=\frac12$2 such that every nontrivial zero satisfies

$\Re(\rho)=\frac12$3

The proof as presented yields

$\Re(\rho)=\frac12$4

while a more careful kernel is said to improve this to $\Re(\rho)=\frac12$5 (Bhowmik et al., 2017).

The argument proceeds through the approximation

$\Re(\rho)=\frac12$6

followed by a square-root estimate

$\Re(\rho)=\frac12$7

A Cauchy integral with kernel

$\Re(\rho)=\frac12$8

then gives

$\Re(\rho)=\frac12$9

The explicit formula for $\Re(s)>1$0 associates a zero $\Re(s)>1$1 with an oscillatory contribution of size approximately $\Re(s)>1$2. A power-saving estimate for $\Re(s)>1$3 therefore excludes zeros with sufficiently large real part.

The logical content is

$\Re(s)>1$4

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(s)>1$5. Consequently, the literal condition $\Re(s)>1$6 should not be identified with RH.

The 2022 paper attempts to strengthen this weak assertion by proving QRH $\Re(s)>1$7 RH. It defines

$\Re(s)>1$8

for zeros $\Re(s)>1$9 and constructs finite Möbius polynomials

$$\zeta(s)=\sum_{n=1}^{\infty}n^{-s} =\prod_p(1-p^{-s})^{-1},$$0

A zero $$\zeta(s)=\sum_{n=1}^{\infty}n^{-s} =\prod_p(1-p^{-s})^{-1},$$1 of $$\zeta(s)=\sum_{n=1}^{\infty}n^{-s} =\prod_p(1-p^{-s})^{-1},$$2 near $$\zeta(s)=\sum_{n=1}^{\infty}n^{-s} =\prod_p(1-p^{-s})^{-1},$$3 is used to form

$$\zeta(s)=\sum_{n=1}^{\infty}n^{-s} =\prod_p(1-p^{-s})^{-1},$$4

which cancels the pole of $$\zeta(s)=\sum_{n=1}^{\infty}n^{-s} =\prod_p(1-p^{-s})^{-1},$$5 at $$\zeta(s)=\sum_{n=1}^{\infty}n^{-s} =\prod_p(1-p^{-s})^{-1},$$6 while retaining the zeta zeros. Symmetrization around a selected large zero produces

$$\zeta(s)=\sum_{n=1}^{\infty}n^{-s} =\prod_p(1-p^{-s})^{-1},$$7

A high-order Taylor expansion, together with estimates for the derivatives of $$\zeta(s)=\sum_{n=1}^{\infty}n^{-s} =\prod_p(1-p^{-s})^{-1},$$8, is then claimed to contradict the existence of a zero with $$\zeta(s)=\sum_{n=1}^{\infty}n^{-s} =\prod_p(1-p^{-s})^{-1},$$9 under the assumption $\mathbb C$0 (Puglisi, 2022). The result is conditional: it does not establish QRH itself.

3. Uniform QRH for families of $\mathbb C$1-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 $\mathbb C$2 such that

$\mathbb C$3

for every primitive finite-order Hecke character $\mathbb C$4 over

$\mathbb C$5

with arbitrary conductor. The same boundary is transferred to Dirichlet $\mathbb C$6-functions and hence to $\mathbb C$7 (Liu, 8 Oct 2026).

The family-wide extremal quantity is

$\mathbb C$8

A bound $\mathbb C$9 means that every member of the family is zero-free in the open half-plane $s=1$0; no assertion is made about the boundary line $s=1$1.

The paper records three successive bounds: $s=1$2 then

$s=1$3

and finally

$s=1$4

The underlying method uses a cubic metaplectic theta average, Poisson summation, reciprocal Hecke $s=1$5-functions, selected prime modifications, character-polynomial detectors, moment estimates, and a Mellin-continuation contradiction. The scale parameters are

$s=1$6

with selected prime lengths satisfying

$s=1$7

The comparison boundary is

$s=1$8

The rational improvement fixes

$s=1$9

giving

$0<\Re(s)<1,$0

The optimized improvement varies both $0<\Re(s)<1,$1 and $0<\Re(s)<1,$2 and produces the radical $0<\Re(s)<1,$3 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 (Liu, 8 Oct 2026).

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,

$0<\Re(s)<1,$4

is compatible with zeros having real parts strictly greater than $0<\Re(s)<1,$5.

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 $0<\Re(s)<1,$6 (Vepstas, 2011).

The classical Gauss-map representation is

$0<\Re(s)<1,$7

where

$0<\Re(s)<1,$8

If

$0<\Re(s)<1,$9

then

$$\rho\longmapsto 1-\rho,\qquad \rho\longmapsto\overline{\rho}.$$0

so $$\rho\longmapsto 1-\rho,\qquad \rho\longmapsto\overline{\rho}.$$1 is the continued-fraction shift map.

The proposed replacement acts on the continued-fraction digits. For positive integers $$\rho\longmapsto 1-\rho,\qquad \rho\longmapsto\overline{\rho}.$$2,

$$\rho\longmapsto 1-\rho,\qquad \rho\longmapsto\overline{\rho}.$$3

The sequence space is represented as

$$\rho\longmapsto 1-\rho,\qquad \rho\longmapsto\overline{\rho}.$$4

with a quotient

$$\rho\longmapsto 1-\rho,\qquad \rho\longmapsto\overline{\rho}.$$5

introduced to handle nonunique continued-fraction expansions.

The associated zeta-like function is

$$\rho\longmapsto 1-\rho,\qquad \rho\longmapsto\overline{\rho}.$$6

The proposed quasi-Riemann-type assertion is

$$\rho\longmapsto 1-\rho,\qquad \rho\longmapsto\overline{\rho}.$$7

for the relevant nontrivial zeros.

The maps $$\rho\longmapsto 1-\rho,\qquad \rho\longmapsto\overline{\rho}.$$8 are stated to map the unit interval onto itself, to have countably many discontinuities, and to be piecewise Möbius: $$\rho\longmapsto 1-\rho,\qquad \rho\longmapsto\overline{\rho}.$$9 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: $\rho$00 The paper studies $\rho$01 in regions with approximately

$\rho$02

using values such as

$\rho$03

The finite sums exhibit artifact zeros. As $\rho$04 increases, some features approach the critical line while other features fade or acquire weak visual intensity. For $\rho$05, the dominant features approximately correspond to those of the Riemann zeta function, although the apparent zeros are visibly shifted. For $\rho$06 and $\rho$07, the correspondence is less clear because the discontinuity structure is finer than the numerical grid resolves.

The paper estimates the first apparent zero of $\rho$08 near

$\rho$09

Its location fluctuates strongly with $\rho$10, and the real part is substantially less stable than the imaginary part. A possible oscillatory component with period approximately

$\rho$11

and an approximately $\rho$12-type power spectrum are reported.

A calibration with $\rho$13 gives

$\rho$14

and

$\rho$15

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 $\rho$16. The critical line is imported by analogy with $\rho$17 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 $\rho$18-function, one normalization is

$\rho$19

Defining

$\rho$20

RH is equivalent to all zeros of $\rho$21 being negative real numbers.

The two-variable $\rho$22-Borel transform is

$\rho$23

The paper proves that for $\rho$24, with

$\rho$25

the zeros of $\rho$26 are represented by convergent power series with real coefficients. For real $\rho$27, the zeros are therefore real.

It defines a threshold $\rho$28 by

$\rho$29

The main equivalence is

$\rho$30

The paper also proves the unconditional bounds

$\rho$31

For Jensen polynomials

$\rho$32

the discriminant criterion is

$\rho$33

RH implies that the zeros of $\rho$34 are simple for $\rho$35, while the converse based solely on simplicity is conjectural (Brugidou, 2012).

5.2 Probabilistic divisibility criteria

For real $\rho$36, define

$\rho$37

This is a characteristic function for every real $\rho$38, with density derived from a theta-function integral. The resulting probability law is called the completed Riemann zeta distribution.

For $\rho$39, RH is equivalent to the assertion that $\rho$40 is “pretended-infinitely divisible,” meaning that it has an exponential Lévy–Khintchine-type representation with a signed measure, without requiring the usual total-variation integrability condition: $\rho$41 Under RH, the relevant signed measure is formally

$\rho$42

Conversely, an exponential representation cannot vanish on the real axis. If $\rho$43 with $\rho$44, choosing $\rho$45 gives

$\rho$46

contradicting such a representation.

For $\rho$47, the paper proves that $\rho$48 is quasi-infinitely divisible but not infinitely divisible: the associated Lévy measure is signed, although it satisfies the total-variation integrability condition (Nakamura, 2015).

5.3 Positivity criteria and generalized Li sums

For real $\rho$49, generalized Li sums are

$\rho$50

The generalized Bombieri–Lagarias criterion gives

$\rho$51

for each admissible fixed $\rho$52, with equivalent formulations using the entire parameter family.

Under RH, the asymptotic is

$\rho$53

The parameter changes the positivity functional and its growth scale, but the zero-location assertion remains the ordinary RH (Sekatskii, 2014).

5.4 Colombeau–Beurling formulations

The Nyman–Beurling and Baez–Duarte criteria characterize RH through the closure of the Beurling functions

$\rho$54

The Baez–Duarte criterion is

$\rho$55

where $\rho$56.

The Colombeau construction introduces damped and truncated Möbius–Beurling nets

$\rho$57

Two damping schemes are given.

For exponential damping,

$\rho$58

with super-exponential truncation

$\rho$59

For polynomial damping,

$\rho$60

$\rho$61

The net is placed in the Colombeau algebra

$\rho$62

and the relevant convergence notion is association: $\rho$63 The paper establishes the equivalence

$\rho$64

This is an alternative generalized-function encoding of RH, not a weaker assertion about zero locations (Cruz et al., 21 Jun 2026).

6. Geometric, analytic, and disputed approaches

Other papers present RH-equivalent inequalities, harmonic reformulations, or claimed proofs whose mathematical status differs sharply.

A modulus representation of $\rho$65 yields

$\rho$66

and a further representation

$\rho$67

Consequently, RH is equivalent to the strict positivity of an explicit cosine-transform expression for every $\rho$68 and $\rho$69. Polynomial approximations and Brownian-motion survival amplitudes provide additional equivalent formulations. These are exact reformulations; they do not themselves prove RH (Sun, 2023).

A harmonic-function approach writes, after shifting $\rho$70,

$\rho$71

The functions $\rho$72 and $\rho$73 are harmonic conjugates: $\rho$74 Zeros of $\rho$75 are intersections of the contours

$\rho$76

The critical line becomes $\rho$77, where symmetry gives $\rho$78. The proposed semi-infinite-strip boundary-value problem, Fourier expansions, and contour nonintersection arguments provide geometric intuition but do not establish the global contour behavior required to prove RH (Nadim, 29 May 2026).

Several papers claim proofs of RH or disproofs of RH, but the supplied analyses identify decisive gaps. The quantum-complex proof uses $\rho$79, so its “quantum” category reduces to ordinary complex manifolds; the proposed rational map on the Riemann sphere is not shown to preserve the zero set of $\rho$80, and the connecting maps are non-holomorphic (Prástaro, 2013). The second-mean-value proof treats a nonunique real mean-value parameter as a globally holomorphic function and applies unjustified operator inversions (Wünsche, 2016). The angular-momentum approach derives the formal identity

$\rho$81

but its prime-product positivity argument is not established on the critical line or throughout the critical strip (Lodone, 2024). The purported disproof based on reciprocal sums over zeros relies on inadequately regularized infinite products, unjustified rearrangements, uncontrolled logarithmic expansions, and numerical inequalities without certified error bounds (Liu, 2024).

The continued-fraction construction likewise remains conjectural and numerical. Its finite approximations can display Riemann-like features even when the limiting function is elementary, demonstrating that visual zero correspondence is not sufficient evidence for exact sharing of zero sets (Vepstas, 2011).

The mathematically established status of “Quasi-Riemann Hypothesis” therefore depends on the usage:

• As $\rho$82, it is a weak zero-free assertion and is not equivalent to RH merely by definition.
• As a uniform bound $\rho$83 for families of $\rho$84-functions, it denotes a quantitative zero-free half-plane.
• For continued-fraction permutation zeta functions, it denotes a conjectural RH-type statement for modified functions.
• In generalized-function, probabilistic, Li-sum, polynomial, and transform settings, it denotes an alternative formulation equivalent to the classical RH.
• In claimed proof or disproof papers, the term has no independently established mathematical consequence unless the analytic gaps are resolved.

Accordingly, no single theorem currently identified as the Quasi-Riemann Hypothesis replaces the classical Riemann Hypothesis. The phrase is best treated as a context-dependent label for weakened zero-free conditions, generalized RH-type constructions, or equivalent reformulations of the classical critical-line assertion.