gpt 6 astra pro attempt on Math-13

Generalized Riemann Hypothesis for Dirichlet L-functions

Definitions and mathematical statement

A Dirichlet character modulo $q$ is a homomorphism $({\mathbb{Z}}/q{\mathbb{Z}})^\times\to{\mathbb{C}}^\times$, extended periodically to integers and by zero on nonunits. Initially, for $\Re s>1$, \[ L(s,\chi)=\sum_{n\ge1}\frac{\chi(n)}{n^s}=\prod_p(1-\chi(p)p^{-s})^{-1}. \] Take the meromorphic continuation. For every $q\ge1$ and every character, ask whether \[ L(\rho,\chi)=0,\quad0<\Re\rho<1\quad\Longrightarrow\quad\Re\rho=\tfrac12. \] Principal and imprimitive characters remain in scope. The restriction to the open strip excludes poles and zeros on its boundary caused by removed Euler factors.

Formulation and concept references: [S1].

Short English statement

Do all Dirichlet $L$-functions place their zeros inside the critical strip on its central vertical line?

Research attempt

Current frontier.

Research check: 27 September 2026. The primitive functional equation, induced-character Euler factors and open-strip zero conventions are recorded in DLMF Section 25.15. [S1]. The August 2026 paper on large quadratic character sums assumes GRH for the Omega results described in its abstract; it cannot be used as an unconditional proof of the assumption. [S2]. We pursue cancellation in the coefficients of $1/L$, rather than positivity of a product of $L$-functions. The integral criterion below is a sufficient condition whose implication is proved here; no new unconditional cancellation estimate is claimed.

Attack route.

A spectral model would need a self-adjoint operator with the actual completed $L$-function as its determinant. A direct character-sum argument controls $L$ but not necessarily its reciprocal. The chosen route is to seek a mean-square estimate for Mobius-twisted sums; it would give a holomorphic reciprocal across the right half of the critical strip. The missing lemma is quantitative cancellation in those sums, with careful attention to the principal and imprimitive cases.

Attempt: reduce to primitive characters without losing cases.

Let $\chi\bmod q$ be induced by a primitive character $\chi^*$ of conductor $q^*\mid q$. Euler products in $\Re s>1$ give \[ L(s,\chi)=L(s,\chi^*) \prod_{\substack{p\mid q\\p\nmid q^*}} (1-\chi^*(p)p^{-s}). \tag{9.1} \] Meromorphic continuation preserves this finite-product identity. For a prime in the product, $|\chi^*(p)|=1$, so a zero of the Euler factor satisfies $p^{-\Re s}=1$, hence $\Re s=0$. Therefore these factors create and cancel no open-strip zeros. The principal character reduces to the primitive character of conductor one, whose $L$-function is $\zeta(s)$; its pole at $s=1$ is not a zero. This is exactly the induced-character convention in [S1], equations 25.15.1--25.15.5. Proving the assertion for primitive characters, including conductor one, suffices for every character in the entry.

Attempt: an $L^2$ criterion for a holomorphic reciprocal.

For a fixed primitive $\chi$ set \[ M_\chi(x)=\sum_{1\leq n\leq x}\mu(n)\chi(n), \] where $\mu(n)=0$ when a square prime factor divides $n$, and $\mu(n)=(-1)^r$ when $n$ is a product of $r$ distinct primes. Consider the following proposed estimate, separately for every $\chi$: \[ \text{for every }\varepsilon>0,\qquad I_{\chi,\varepsilon}:= \int_1^\infty |M_\chi(x)|^2x^{-2-\varepsilon}\,dx<\infty. \tag{9.2} \] Constants may depend on $\chi$ and $\varepsilon$. A uniform bound over all conductors is not added to this sufficient condition.

Assuming $9.2$, define \[ H_\chi(s)=s\int_1^\infty M_\chi(x)x^{-s-1}\,dx \quad(\Re s>1/2). \tag{9.3} \] To justify this definition, write $\sigma=\Re s$ and choose $0<\varepsilon<2\sigma-1$. Cauchy--Schwarz gives \[ \int_1^\infty |M_\chi(x)|x^{-\sigma-1}\,dx \leq I_{\chi,\varepsilon}^{1/2} \left(\int_1^\infty x^{-2\sigma+\varepsilon}\,dx\right)^{1/2} =\frac{I_{\chi,\varepsilon}^{1/2}}{ \sqrt{2\sigma-\varepsilon-1}}. \tag{9.4} \] For a compact subset of the half-plane, choose a single $\varepsilon$ smaller than twice its distance from the boundary. The same argument, with $(\log x)^{2j}$ in the second integral, bounds every derivative of the integrand. Consequently $9.3$ is holomorphic, by locally uniform convergence and differentiation under the integral. There is no unjustified interchange at $\Re s=1/2$; that boundary is not included.

In $\Re s>1$, the absolutely convergent Euler products give \[ H_\chi(s)=\sum_{n\geq1}\frac{\mu(n)\chi(n)}{n^s} =\frac1{L(s,\chi)}. \tag{9.5} \] The first equality is summation by parts; $M_\chi(x)=O(x)$ is enough for its endpoint term to vanish there. Since $L$ has the continuation specified in the original statement, meromorphic identity extends $L(s,\chi)H_\chi(s)=1$ to $\Re s>1/2$. A zero of $L$ at a point where it is holomorphic contradicts this identity. At $s=1$ for conductor one, $H_\chi$ instead has a zero cancelling the pole, which is harmless. The primitive functional equation relates a zero $\rho$ to a zero $1-\rho$ of the conjugate character's completed function. Gamma factors are finite and nonzero in the open strip. Applying the right-half-plane argument to all primitive characters, including their conjugates, therefore excludes left-half-strip zeros as well. [S1]. Together with $9.1$, this proves the original GRH assertion under $9.2$.

Attempt: a dyadic estimate would close the analytic step.

It would suffice to show, for every $\varepsilon>0$ and $X\geq1$, \[ \int_X^{2X}|M_\chi(x)|^2\,dx \leq C_{\chi,\varepsilon}X^{2+\varepsilon/2}. \tag{9.6} \] Indeed the contribution of $[2^j,2^{j+1}]$ to $9.2$ is at most $C_{\chi,\varepsilon}2^{-j\varepsilon/2}$, a convergent geometric series. This mean-square condition allows spikes and is not silently replaced by a pointwise bound. The trivial estimate $|M_\chi(x)|\leq x$ gives a block integral of order $X^3$, which misses the needed exponent by nearly one. The integer-squarefree weight $\mu$ is thus the substantive cancellation issue; the Mellin analysis by itself does not improve it.

Stress test: cancellation of $\chi$ is not cancellation of $\mu\chi$.

For a nonprincipal character modulo $q$, summing over a complete period gives zero. Thus $\sum_{n\leq x}\chi(n)=O(q)$, yielding a direct continuation formula for $L(s,\chi)$ on $\Re s>0$. There is no corresponding periodicity for $\mu(n)\chi(n)$: divisibility by squares and parity of the number of prime factors are not periodic modulo $q$. Replacing $M_\chi$ by the ordinary character sum would prove continuation of $L$, not continuation of its reciprocal.

A small analytic countermodel pinpoints that logical difference. The Dirichlet polynomial \[ D(s)=1-\tfrac32\,2^{-s} \] has bounded coefficient partial sums and is entire, but vanishes at $\beta=\log(3/2)/\log2$, with $1/2<\beta<1$. Its reciprocal has a pole there. This polynomial is not asserted to be a Dirichlet $L$-function; it disproves only the inference from bounded coefficient sums and continuation to zero-freeness. Likewise, agreement of numerical partial sums with square-root growth up to a finite cutoff would not establish $9.2$, whose integral runs to infinity.

Outcome.

Outcome: Conditional reduction. The attempt gives a fully justified sufficient mean-square criterion, with explicit compact convergence, principal-character treatment, and finite-Euler-factor bookkeeping. A proof of $9.2$, or the stronger dyadic bound $9.6$, for every primitive character would settle the exact entry. Neither estimate is established here, and the argument does not claim a converse or historical novelty for the reciprocal-Mellin method. The obstruction is arithmetic cancellation, not an omitted analytic continuation step.

Sources

• S1 NIST DLMF25.15, Dirichlet L-functions, equations25.15.1–4 and zeros; GRH convention in Kandhil–Languasco–Moree, Mathematische Annalen394:43$2026$. https://dlmf.nist.gov/25.15. Catalogue formulation source. Consulted 24 September 2026: Dirichlet characters, L-functions and zeros.
• S2 Large values of quadratic character sums — Dong, Wang, Wang, Zhang and Zhao. https://arxiv.org/abs/2608.15773. Catalogue research/status reference; not a proof endorsement.
• S3 On the Extended, Generalized, and Grand Riemann Hypotheses: A Unified Approach Based on the General Properties of L-Functions — Weicun Zhang, v19. https://www.preprints.org/manuscript/202506.0481. Catalogue research/status reference; not a proof endorsement.