gpt 6 astra ultra attempt on Math-139

Selberg's 1/4 conjecture

Definitions and mathematical statement

The upper half-plane $\mathbb H=\{x+iy:y>0\}$ has hyperbolic metric $y^{-2}(dx^2+dy^2)$ and nonnegative Laplacian $\Delta=-y^2(\partial_x^2+\partial_y^2)$. A congruence subgroup contains a principal subgroup $\Gamma(N)$ for some $N$. On its finite-area quotient, the assertion excludes any nonconstant $L^2$ eigenfunction with \[ \Delta f=\lambda f,\qquad0<\lambda<\tfrac14. \] This formulation remains meaningful for noncompact quotients, where a continuous spectrum is also present; "first nonzero eigenvalue" is understood as the bottom of the nonconstant spectral range when necessary.

Formulation and concept references: [S1].

Short English statement

Do all congruence hyperbolic surfaces have a spectral gap of at least one quarter above the constant functions?

Research attempt

Current frontier.

Write a possible exceptional eigenvalue as $\lambda=1/4-\nu^2$, with $0<\nu<1/2$. The uniform Kim--Sarnak bound is $\nu\le 7/64$, or $\lambda\ge 975/4096$; Pascadi's January 2026 revision explicitly records this bound in Theorem C. His improved large-sieve estimates control certain weighted contributions of exceptional forms, rather than proving that exceptional forms do not exist. [E1] Booker, Lee, and Strömbergsson prove the conjecture for $\Gamma_1(N)$, $N\le880$, and $\Gamma(N)$, $N\le226$, in Theorem 1.1 of [S3]. Neither a finite level range nor a small average contribution excludes one eigenfunction at an arbitrary remaining level.

The analytic framework used below is the cofinite pre-trace formula, including the Eisenstein integral, not the trace of an unregularized heat operator on a noncompact surface. The transform convention and local spectral estimates are those recalled in Section 2 of [E2]. The explicit filter and the conditional argument that follows are deductions of this notebook; no novelty claim is made for the general principle of detecting exceptional spectrum by positive test functions.

Attack route.

Three routes were compared. Symmetric-power amplification would need suitable lifts in arbitrarily large degree for the particular Maa

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representation: if a lift detects $m\nu$ and bounds it by a constant independent of $m$, then $\nu=0$. This describes a missing functoriality input, not a theorem being invoked. A level-density argument would need to exclude a single exceptional representation, not merely show that exceptions have density zero. The route pursued here instead constructs a spectral filter that kills the constant function exactly, remains nonnegative on every other spectral component, and has compact geometric support. Its missing input will be a concrete signed orbit-counting estimate.

Attempt: reduction to a fixed principal quotient.

It suffices to handle the effective image of $\Gamma(N)$ in $\operatorname{PSL}_2({\mathbb{R}})$ for every $N\ge3$. Indeed, any congruence subgroup contains such a principal subgroup after replacing its level by a sufficiently large multiple. An exceptional eigenfunction pulls back to the finite cover; its eigenvalue is unchanged and its squared norm is multiplied by the degree of the cover. The pullback is still nonconstant and square integrable. Thus fix a torsion-free principal group $\Gamma$, put $Y=\Gamma\backslash\mathbb H$, and write $V=\operatorname{vol}(Y)$. Constants below may depend on this fixed quotient. No uniform-in-level constant will be silently assumed.

Choose $\phi\in C_c^\infty(Y)$, $\phi\ge0$, positive on a nonempty open set, and put $\Phi=\int_Y\phi\,d\mu$. Let $u_j$ be an orthonormal list of discrete $L^2$ eigenfunctions, including the constant and any residual eigenfunctions, with $\lambda_j=1/4+r_j^2$. The parameter $r_j$ is real and nonnegative, or lies on $i(0,1/2]$. For $T\ge1$, set \begin{equation}\label{wsref-0-72-61-39-34-30-30-30-37-39-3a-66-69-6c-74-65-72} h_T(r)=\left(r^2+\frac14\right)^2 \left(\frac{\sin(Tr)}{Tr}\right)^8, \qquad \frac{\sin 0}{0}:=1. \end{equation} This is entire and even. For real $r$ it is nonnegative. Moreover, \begin{equation}\label{wsref-0-72-61-39-34-30-30-30-37-39-3a-65-78-63-65-70-74-69-6f-6e} h_T(i/2)=0,\qquad h_T(i\nu)=\left(\frac14-\nu^2\right)^2 \left(\frac{\sinh(T\nu)}{T\nu}\right)^8>0 \quad(0<\nu<1/2). \end{equation} The zero at $i/2$ removes the constant without subtracting a large positive quantity numerically. The eighth power provides enough decay on the real spectral axis after multiplication by the degree-four factor.

A localized positive quantity.

Let $k_T(s)$ be the radial inverse Selberg transform of $h_T$, where $s$ is hyperbolic distance. Define \begin{equation}\label{wsref-0-72-61-39-34-30-30-30-37-39-3a-71-75-61-6e-74-69-74-79} Q_{\Gamma,\phi}(T)= \int_Y\phi(z)\sum_{\gamma\in\Gamma} k_T\left(d(z,\gamma z)\right)\,d\mu(z). \end{equation} The cofinite pre-trace formula, specialized to this filter, gives \begin{align}\label{wsref-0-72-61-39-34-30-30-30-37-39-3a-70-72-65-74-72-61-63-65} Q_{\Gamma,\phi}(T) &=\sum_j h_T(r_j)\int_Y\phi|u_j|^2\,d\mu \\ &\quad+\frac1{4\pi}\sum_{\mathfrak a} \int_{{\mathbb{R}}}h_T(r) \left(\int_Y\phi(z) |E_{\mathfrak a}(z,1/2+ir)|^2\,d\mu(z)\right)dr.\notag \end{align} Here the sum over $\mathfrak a$ runs over the cusps, with the usual Eisenstein normalization. This use of the spectral formula and its continuous term is supported by [E2], Section 2, especially equation $2.2$.

For completeness, the needed convergence is not a formal interchange. The filter is $O_T((1+|r|)^{-4})$ in a fixed horizontal strip. On the compact support of $\phi$, the cumulative local spectral mass, combining the discrete and continuous real spectrum up to $|r|\le R$, is $O_{\Gamma,\phi}(1+R^2)$. Dyadic summation therefore makes the real-spectrum terms absolutely integrable. There are finitely many exceptional eigenvalues. On the geometric side $k_T$ has support in $[0,8T]$, as proved below, so only finitely many orbit terms occur locally, uniformly on this compact support. Thus both sides of (ra9400079:pretrace) are legitimate finite or absolutely convergent expressions.

All terms on the right of (ra9400079:pretrace) are nonnegative. An exceptional $u$ has $b_u:=\int_Y\phi|u|^2\,d\mu>0$: otherwise it vanishes on an open set, and unique continuation for the Laplace equation forces $u=0$. Consequently an exceptional parameter $\nu$ implies, for all sufficiently large $T$, \begin{equation}\label{wsref-0-72-61-39-34-30-30-30-37-39-3a-67-72-6f-77-74-68} Q_{\Gamma,\phi}(T)\ge c_{\Gamma,\phi,u,\nu}\,T^{-8}e^{8\nu T}. \end{equation} Neither the constant nor the onset of this estimate is asserted to be uniform as $\nu\downarrow0$.

Conversely, if no exceptional eigenvalue exists, then $Q_{\Gamma,\phi}(T)$ is bounded independently of $T\ge1$. Indeed, for real $r$, \[ 0\le h_T(r)\le \left(r^2+\frac14\right)^2\min\{1,|r|^{-8}\}, \] where the right side is interpreted by its bounded extension near zero. The same local spectral-mass bound integrates this majorant. In particular, the threshold $r=0$ causes no difficulty: $h_T(0)=1/16$.

Filter lemma. For this fixed $\Gamma$ and $\phi$, absence of exceptional $L^2$ eigenvalues is equivalent to \begin{equation}\label{wsref-0-72-61-39-34-30-30-30-37-39-3a-73-75-62-65-78-70} \forall\epsilon>0\quad Q_{\Gamma,\phi}(T)\ll_{\Gamma,\phi,\epsilon}e^{\epsilon T} \quad(T\ge1). \end{equation} The forward implication is the boundedness just proved. For the reverse implication, choose $0<\epsilon<8\nu$ in (ra9400079:growth). $\square$

This is a proved reformulation, not an improvement of the spectral gap. Its usefulness depends entirely on estimating (ra9400079:quantity) by arithmetic geometry without first assuming the desired spectral conclusion.

An explicit geometric kernel and a sufficient missing estimate.

The Fourier convention is $h(r)=\int_{{\mathbb{R}}}g(u)e^{iru}\,du$. Put \[ b_T(u)=\frac{\mathbf 1_{[-T,T]}(u)}{2T},\qquad B_T=b_T^{*8},\qquad g_T=\left(-\frac{d^2}{du^2}+\frac14\right)^2B_T. \] Then $\widehat{B_T}(r)=(\sin(Tr)/(Tr))^8$, so $g_T$ transforms to $h_T$. The function $B_T$ is a compactly supported piecewise polynomial of class $C^6$, and $g_T$ is of class $C^2$, supported in $[-8T,8T]$. Abel inversion of the transform convention in [E2], Section 2, gives, now expressed in the distance variable, \begin{equation}\label{wsref-0-72-61-39-34-30-30-30-37-39-3a-69-6e-76-65-72-73-65} k_T(s)=-\frac1{\sqrt2\,\pi} \int_s^\infty\frac{g_T'(u)}{\sqrt{\cosh u-\cosh s}}\,du. \end{equation} The square-root singularity is integrable for $s>0$; at $s=0$, evenness gives $g_T'(u)=O_T(u)$. This formula also proves the stated support property.

Here are bounds adequate for testing the route. Scaling gives $B_T(u)=T^{-1}B_1(u/T)$, hence $\|g_T'\|_\infty\ll T^{-2}$ and $\|g_T''\|_\infty\ll T^{-3}$ for $T\ge1$. For $s\ge1$, substitute $u=s+v$ in (ra9400079:inverse) and use \[ \cosh(s+v)-\cosh s =2\sinh(v/2)\sinh(s+v/2). \] The resulting denominator has an integrable $v^{-1/2}$ singularity at zero and exponential decay after inversion at infinity. Its logarithmic $s$-derivative is bounded for $s\ge1$. Differentiation under this dominated integral is therefore allowed and yields \begin{equation}\label{wsref-0-72-61-39-34-30-30-30-37-39-3a-6b-65-72-6e-65-6c-62-6f-75-6e-64} |k_T(s)|+|k_T'(s)|\ll T^{-2}e^{-s/2}\quad(s\ge1). \end{equation} For $0\le s\le1$, use $|g_T'(u)|\le\|g_T''\|_\infty u$ for $u\le2$, and the exponential bound for the remaining integral; this gives $\sup_{[0,1]}|k_T|\ll1$. No derivative bound at the origin is needed below.

Consider the locally averaged orbit count \[ N_\phi(R)=\int_Y\phi(z)\, \#\{\gamma\in\Gamma:d(z,\gamma z)\le R\}\,d\mu(z) \] and the discrepancy with only the constant eigenfunction's main term removed: \begin{equation}\label{wsref-0-72-61-39-34-30-30-30-37-39-3a-65-72-72-6f-72} \mathcal E_\phi(R)=N_\phi(R) -\frac{\Phi}{V}\,2\pi(\cosh R-1). \end{equation} Since $2\pi\int_0^\infty k_T(R)\sinh R\,dR=h_T(i/2)=0$, one has the exact Stieltjes identity \begin{equation}\label{wsref-0-72-61-39-34-30-30-30-37-39-3a-73-74-69-65-6c-74-6a-65-73} Q_{\Gamma,\phi}(T)=\int_{[0,8T]}k_T(R)\,d\mathcal E_\phi(R). \end{equation} The atom at $R=0$, coming from the identity element, is included.

A sufficient arithmetic input would be the following statement: \begin{equation}\label{wsref-0-72-61-39-34-30-30-30-37-39-3a-6d-69-73-73-69-6e-67} \boxed{\quad \forall\epsilon>0\qquad |\mathcal E_\phi(R)|\ll_{\Gamma,\phi,\epsilon} e^{(1/2+\epsilon)R}\quad(R\ge1). \quad} \end{equation} To check the implication carefully, the part of (ra9400079:stieltjes) on $[0,1]$ is $O_{\Gamma,\phi}(1)$ because the counting measure has finite mass there and $k_T$ is uniformly bounded. On $[1,8T]$, integration by parts and (ra9400079:kernelbound) give \[ |Q_{\Gamma,\phi}(T)| \ll_{\Gamma,\phi,\epsilon} 1+T^{-2}\int_1^{8T}e^{\epsilon R}\,dR \ll_{\Gamma,\phi,\epsilon}1+e^{8\epsilon T}. \] The outer endpoint term vanishes because $k_T(8T)=0$; the inner endpoint is bounded. Choosing $\epsilon$ arbitrarily small proves (ra9400079:subexp). Thus (ra9400079:missing), for every principal quotient, would imply exactly the conjecture in this entry, including possible residual spectrum and noncompact quotients.

Stress test.

The indispensable sign cancellation. It is tempting to discard signs in the orbit sum. That destroys the argument. A packing bound on compact subsets gives $N_\phi(R)\ll_{\Gamma,\phi}e^R$. Combining this with $|k_T(R)|\ll T^{-2}e^{-R/2}$ gives at best an exponential bound of order $e^{4T}$, up to polynomial factors. Comparison with (ra9400079:growth) recovers only the trivial restriction $\nu\le1/2$. Furthermore, $k_T$ cannot be nonnegative everywhere: its hyperbolic integral is zero, while its transform is not identically zero. Spectral positivity and geometric positivity are different assertions.

A circular subtraction to avoid. In the usual hyperbolic circle problem the main term often includes explicit contributions from all small eigenvalues. This is precisely the convention in [E2], equation $1.2$. A square-root-size error after subtracting those terms would not prove Selberg. Our (ra9400079:error) deliberately does not subtract them. Estimate (ra9400079:missing) is therefore stronger than a remainder estimate that already allows exceptional spectrum; it has not been proved here.

Why an average or a limiting argument does not close the gap. An exceptional eigenfunction on one quotient persists on every finite cover. Dividing a spectral count by the growing covering degree can nevertheless make this fixed contribution invisible. Thus a density-zero statement in a tower is compatible with the very counterexample being sought. Here $\Gamma$ and $\phi$ are fixed before $T\to\infty$. Likewise, applying the known $\nu\le7/64$ bound to the positive spectral side only gives a bound with exponential rate at most $8(7/64)=7/8$; feeding that bound back into (ra9400079:growth) reproduces $7/64$, not a smaller number. [E1]

Dependency check. Improved exceptional-spectrum estimates can feed into the distribution of prime factors of polynomial values, as Pascadi's application demonstrates. They do not by themselves distinguish a prime value from a product of two primes. Thus neither this conditional route nor the existing weighted estimates establish the Bunyakovsky assertion in Section 1292. [E1]

Outcome.

Outcome: Conditional reduction.

The explicit filter, the localized positivity lemma, and the implication from (ra9400079:missing) have been justified. The missing estimate is stated with its exact main term, quantifiers, and dependence on the quotient. The attempt also identifies the exponential loss caused by taking absolute values and the circularity of subtracting unknown exceptional terms.

No new numerical lower bound for $\lambda_1$ has been obtained. A complete resolution by this route requires an independent arithmetic proof of (ra9400079:subexp), or of the stronger orbit discrepancy estimate (ra9400079:missing), on every principal quotient. Positivity alone does not supply that proof.

Sources

• S1 Formal statement, Chapter 4, Selberg property. https://www.ma.huji.ac.il/~alexlub/BOOKS/On%20property/On%20property.pdf. Catalogue formulation source.
• S2 Asymptotic independence for random permutations from surface groups. https://link.springer.com/article/10.1007/s10711-025-01003-8. Catalogue research/status reference; not a proof endorsement.
• S3 Twist-minimal trace formulas and the Selberg eigenvalue conjecture. https://arxiv.org/abs/1803.06016. Catalogue research/status reference; not a proof endorsement.
• E1 A. Pascadi, Large sieve inequalities for exceptional Maass forms and the greatest prime factor of $n^2+1$, Forum of Mathematics, Pi 14 $2026$, e8; arXiv:2404.04239v3, 15 January 2026. Introduction and Theorem C (the Kim--Sarnak $7/64$ bound), and the exceptional-spectrum large-sieve results. https://arxiv.org/html/2404.04239v3. https://doi.org/10.1017/fmp.2026.10025. Consulted 25 September 2026; weighted estimates are not an exclusion of exceptional eigenvalues.
• E2 G. Cherubini and M. S. Risager, On the variance of the error term in the hyperbolic circle problem, Revista Matemática Iberoamericana 34 $2018$, no. 2, 655--685, DOI 10.4171/RMI/1000. Section 2, transform convention and pre-trace formula $2.2$; equation $1.2$ for the main term including small eigenvalues; Section 4 for smoothing and absolute convergence. https://ems.press/content/serial-article-files/38698. Consulted 25 September 2026.