A fixed zero-free region for zeta and Dirichlet L-functions

← All digests

Family 003 · Number theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

Prime numbers become less frequent as numbers grow, but their local distribution is irregular. One way to study that irregularity is to replace the primes by a complex function whose zeros encode fluctuations in prime counts. The Riemann hypothesis predicts an extraordinarily precise location for those zeros. A weaker question has also resisted solution: can they at least be kept a fixed positive distance away from the vertical line with real part one?

Family 003 claims an affirmative answer. Its strongest manuscript, M1, states that the Riemann zeta function and every Dirichlet L-function have no zeros in

$$\operatorname{Re}s>\frac78.$$

The same assertion covers finite-order Hecke L-functions over one particular quadratic number field. M2 supplies a separate presentation of a weaker fixed boundary, $11/12$, and several arithmetic consequences. M3 addresses the exceptional real zeros called Landau–Siegel zeros through a different, algebraic argument.

The distinction between these claims and the Riemann hypothesis is substantial. Moving the forbidden boundary from one to seven-eighths would be a major advance, but the conjectured boundary is one-half. This digest reports the manuscripts’ claims; it does not establish their correctness.

2. Background and essential definitions

For a complex number $s=\sigma+it$, the number $\sigma$ is its real part and $t$ its height. Initially, for $\sigma>1$, the zeta function is defined by

$$\zeta(s)=\sum_{n=1}^{\infty}\frac1{n^s} =\prod_{p\text{ prime}}(1-p^{-s})^{-1}.$$

The product explains why this analytic function knows about primes. Analytic continuation extends it beyond the region where this series converges, with a pole at $s=1$. A pole is a singularity, not a zero. Its nontrivial zeros lie in the critical strip $0<\sigma<1$; the Riemann hypothesis places them on $\sigma=1/2$.

A Dirichlet character $\chi$ modulo a positive integer $q$ is a periodic, multiplicative function of integers. It is zero on integers sharing a factor with $q$, and takes root-of-unity values on the others. Characters separate residue classes in sums, allowing one to count primes such as those congruent to one or three modulo four. Their functions are

$$L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s} =\prod_p(1-\chi(p)p^{-s})^{-1} \qquad(\sigma>1).$$

The conductor is the smallest modulus from which the character is induced. A character is primitive if its displayed modulus is that conductor. Real nonprincipal characters take values $0,1,-1$; a real zero very close to one is the potential exceptional phenomenon in classical zero-free theorems.

Hecke characters generalize this language from integers to ideals of a number field. Here the field is

$$K=\mathbb Q(\sqrt{-3}),\qquad \mathcal O_K=\mathbb Z[\omega],\qquad \omega=e^{2\pi i/3}.$$

An ideal is an additive subgroup of this ring closed under multiplication by every ring element. Its norm $N\mathfrak a$ is the number of elements in the finite quotient $\mathcal O_K/\mathfrak a$. A finite-order Hecke character $\eta$ assigns roots of unity to ideals, subject to multiplicativity and congruence conditions, and is extended by zero at ideals not coprime to its conductor. Its L-series replaces $n$ by $N\mathfrak a$.

“Fixed zero-free half-plane” means a single boundary works for every height and every character under consideration. Constants in intermediate estimates may depend on a character without changing that common boundary.

3. Development history and previous partial results

Riemann’s 1859 memoir connected prime counting with the zeros of $\zeta$. In 1896, Hadamard and de la Vallée Poussin proved that no zero lies on $\sigma=1$, leading to the prime number theorem: the number of primes up to $x$ is asymptotic to $x/\log x$. Hadamard’s original paper makes that connection explicit.

Later work pushed the zero-free region slightly left of one. A classical Dirichlet L-function theorem has a boundary of the approximate shape

$$\sigma>1-\frac{c}{\log(q(|t|+3))},$$

with an absolute positive constant $c$, apart from a possible simple real zero for a real character. The permitted width shrinks as conductor or height grows. Vinogradov–Korobov estimates improve the height dependence for zeta; Ford’s account and explicit estimates develop that direction. Their boundary still approaches one at large height. A fixed strip is a qualitatively different conclusion.

The possible real exception has its own history. Siegel’s 1935 theorem gives a strong lower bound for $L(1,\chi)$, but its constant is ineffective: the argument does not provide a computable value. Page’s theorem severely limits how many exceptional characters can occur in a conductor range. Deuring–Heilbronn repulsion says that an exceptional zero pushes other zeros away. These results constrain the obstruction without uniformly eliminating it. M3 situates its claim in this development.

The analytic manuscripts draw on another line: Kubota’s automorphic functions, Patterson’s cubic theta series, and large-sieve inequalities controlling averages of character sums. Dunn and Radziwiłł’s work on cubic Gauss sums supplies unconditional theta-expansion formulas used in this framework. Their conditional prime asymptotic is not an input to M1. The separate unconditional Patterson claim in family 023 is also not an input to M1.

4. What the new manuscripts claim

Three different statements

M1, Theorem 1.1, asserts that every finite-order Hecke L-function over $K=\mathbb Q(\sqrt{-3})$, and every Dirichlet L-function, is zero-free for $\sigma>7/8$. Principal characters may have a pole at one. The boundary line itself is excluded from the assertion.

M2, Theorem 1.1, states the same families are zero-free for $\sigma>11/12$. Since $7/8<11/12$, M1 gives the stronger exclusion. M2 isolates a shorter argument and spells out consequences; these are companion treatments within one family, not three equivalent formulations of RH.

M3 states that there is an absolute $c>0$ such that every real zero $\beta\in(0,1)$ of a primitive nonprincipal real Dirichlet character of conductor $q\ge3$ satisfies

$$(1-\beta)\log q\ge c.$$

This is a uniform logarithmic gap for real zeros. It addresses the exceptional-zero problem but does not itself supply a fixed half-plane for all complex zeros. The stronger conclusions of M1 and M2 overlap its real-zero consequence.

Consequences stated in M2

Let $\pi(x;q,a)$ count primes at most $x$ congruent to $a$ modulo $q$, and let $\varphi(q)$ count the residue classes coprime to $q$. M2 gives

$$\sup_{\substack{1\le q\le x\\(a,q)=1}} \left|\pi(x;q,a)-\frac1{\varphi(q)} \int_2^x\frac{dt}{\log t}\right| \ll x^{11/12}\log x.$$

Here $\ll$ means bounded by an absolute constant times the expression on the right; M2 states that constant is effective. For very large $q$, the error can exceed the main term. Uniformity alone does not make this a useful relative approximation in every progression.

M2 also claims a fixed-power-of-$\log p$ bound for the least quadratic nonresidue modulo an odd prime $p$: the smallest positive integer that is not a nonzero square modulo $p$. Through the Tonelli–Shanks procedure, this gives deterministic square-root extraction in time polynomial in the number of digits of $p$. Its consequence for Miller’s primality test supplies another deterministic implementation; polynomial-time primality testing itself was already known through AKS.

5. Why the advance matters

Provisional importance: 5/5. A uniform fixed zero-free region would change the available unconditional tools across analytic number theory. It addresses a central longstanding obstruction and yields consequences for prime distributions, quadratic fields, and algorithms.

The gain is stronger than a better numerical constant in the classical shrinking region. For example, knowing that no zero lies farther right than seven-eighths controls contributions from zeros by a fixed power of the scale. Classical regions often give savings that diminish with height. That difference propagates into applications.

M3 is important for a complementary reason: it claims to eliminate an obstruction whose possible existence has forced many results to carry an exception or an ineffective constant. The score reflects these mathematical consequences if the claims hold, not an assessment of proof reliability.

6. The main idea

Turn a zero question into a cancellation question

The integer Möbius function $\mu(n)$ is zero if a square of a prime divides $n$, and otherwise equals $(-1)^k$ when $n$ has $k$ distinct prime factors. In the absolutely convergent region,

$$\frac1{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}.$$

Strong cancellation in sums of these signs can continue the reciprocal farther left. A zero of zeta would be a pole of that reciprocal, so analytic continuation with suitable bounds excludes zeros. The same principle works for character twists and the ideal Möbius function.

M2 studies smoothed sums

$$A_1(D)=\sum_{\mathfrak a} \mu(\mathfrak a)\eta(\mathfrak a) W(N\mathfrak a/D),$$

with the manuscript’s coprimality restrictions. The smooth function $W$ selects ideals whose norms are comparable to $D$. A trivial estimate is roughly $D$; the target is $D^{11/12+\varepsilon}$ for every positive $\varepsilon$.

Instead of estimating this one sum directly, M2 inserts sixth-power residue characters and studies a family $A_u(D)$. A mean-square bound says that these sums are small on average. Many indices $u$ that are sixth powers reproduce the original sum up to controlled omissions. Consequently, an oversized original sum would force too many large members of the family.

Poisson summation, a form of Fourier duality, transforms the average. Finite Fourier transforms of residue characters produce Gauss sums. Arithmetic identities combine these with Möbius signs to produce cubic theta coefficients. Transformation laws of the theta function and recursive mean-square estimates then provide the asserted cancellation. Retaining full cube-divisor contributions is part of that argument, not a removable cosmetic detail.

How the stronger boundary is approached

M1 also compares two descriptions of completed cubic-theta sums: a reflected expression controlled by mean-square estimates, and a Poisson expression containing a reciprocal L-function. The first stage reaches $11/12$. A second stage changes the weighting at selected primes and uses unequal averaging scales.

A hypothetical zero produces two large finite sums with the same character and height: one has Möbius coefficients, the other does not. Separate moment estimates bound how often both can be large. The manuscript combines these bounds, while treating principal terms and residues separately, to move the common boundary to $7/8$. The simultaneous treatment of Hecke characters matters even for zeta, because transformations introduce additional twists.

The algebraic route in M3

M3 translates a hypothetical exceptional zero into an imbalance between primes on which a real character takes signs $+1$ and $-1$. It then constructs a nonzero interpolation determinant from algebraic numbers in a biquadratic field, a degree-four extension obtained by adjoining two square roots.

At primes of the plentiful sign, taking a $p$-th power modulo $p$ acts like a particular field conjugation. Row operations therefore force high powers of many primes to divide the determinant’s norm, which is a nonzero integer. A carefully weighted selection of polynomial rows keeps its ordinary size small. The claimed contradiction is between excessive arithmetic divisibility and this size bound. This mechanism differs from the theta-function comparisons in M1 and M2.

7. Scope, limitations, and open questions

The Riemann hypothesis and generalized Riemann hypotheses remain open in these manuscripts: neither boundary reaches one-half. No conclusion is asserted on the line $\sigma=7/8$ itself. The Hecke theorem is stated for finite-order characters of the specified imaginary quadratic field, not arbitrary automorphic L-functions over all number fields.

The three source statements should retain their different strengths. M3’s logarithmic real-zero exclusion is not a substitute for M1’s full fixed half-plane. M2’s algorithmic applications also do not make every algorithm previously conditional on GRH unconditional; each reduction has its own hypotheses.

Further questions include whether the analytic estimates can reach a smaller fixed boundary, whether related methods extend to other fields and higher-degree L-functions, and which applications can make efficient use of the explicit savings. These are directions suggested by the statements, not additional conclusions of this digest.

8. Sources and further reading

M1’s introduction and proof overview explain the two-stage $11/12$-to-$7/8$ strategy. M2’s Theorem 1.1, Corollary 1.2, and Section 2 provide the shorter claim, applications, and Möbius-sum outline. M3’s introduction describes the real-zero theorem and determinant mechanism.

The historical links in Section 3 lead to primary papers. For a systematic background reference, Iwaniec and Kowalski’s Analytic Number Theory develops characters, explicit formulas, and large sieves. The Möbius function is a useful optional entry point to the cancellation viewpoint.

Source manuscripts

• M1. OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8 (30 September 2026).
• M2. OpenAI, The Quasi-Riemann Hypothesis (October 5, 2026).
• M3. OpenAI, Uniform exclusion of Landau–Siegel zeros (October 1, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.