--- title: Selberg Orthogonality Conjecture url: https://www.emergentmind.com/topics/selberg-orthogonality-conjecture type: topic --- # Selberg Orthogonality Conjecture Searching arXiv for the cited papers to ground the article in current literature. The Selberg Orthogonality Conjecture is a family of asymptotic orthogonality statements for automorphic coefficients on $\GL_n$ and $\PGL(N)$. In one formulation, it asserts that Fourier coefficients or Hecke eigenvalues of Maass forms, after localization in spectral parameter and weighting by $1/L(1,\mathrm{Ad})$, become asymptotically orthonormal in a vertical family. In another, it asserts that prime coefficients of distinct cuspidal automorphic representations have only the diagonal $\log\log x$ correlation. These formulations connect the Kuznetsov–Petersson paradigm in rank one to higher-rank trace formulas, Rankin–Selberg theory, and vertical Sato–Tate equidistribution [1303.0889], [2212.14534], [2507.20653]. ## 1. Formulations of the conjecture For Maass forms on $\PGL(N)$, let $\{\phi_j\}$ be an orthonormal Hecke–Maass basis for \[ L^2_\mathrm{cusp}\bigl(SL(N,\Z)\backslash GL(N,\R)/O(N)\bigr), \] and write $A_j(m_1,\dots,m_{N-1})$ for the $(m_1,\dots,m_{N-1})$-th Fourier coefficient of $\phi_j$, normalized so that \[ A_j(1,\dots,1)=1. \] If $v(\phi_j)\in\C^{N-1}$ is the spectral parameter and $h_T(v(\phi))\ge 0$ is a smooth cut-off which for large $T$ essentially restricts to $\|v(\phi)\|\ll T$, Zhou defines the harmonic weight \[ \omega_j(T)=\frac{h_T\bigl(v(\phi_j)\bigr)}{L\bigl(1,\phi_j,\mathrm{Ad}\bigr)}. \] The conjectured orthogonality relation is \[ \lim_{T\to\infty} \frac{\displaystyle \sum_j A_j(m_1,\dots,m_{N-1})\,\overline{A_j(n_1,\dots,n_{N-1})}\;\omega_j(T)} {\displaystyle\sum_j\omega_j(T)} = \delta_{\mathbf m,\mathbf n}, \] where $\mathbf m=(m_1,\dots,m_{N-1})$ and $\mathbf n=(n_1,\dots,n_{N-1})$ are tuples of positive integers, and $\delta_{\mathbf m,\mathbf n}=1$ exactly when $m_i=n_i$ for all $i$ [1303.0889]. Zhou also notes that one can replace the two-sided sum by a one-sided weak form and recover the full form by Hecke relations. A parallel asymptotic orthogonality relation is formulated in [2212.14534] for Hecke eigenvalues of Maass cusp forms on $\SL(n,\Z)$. If \[ L(s,\phi_j)=\sum_{m=1}^\infty \lambda_j(m)m^{-s},\qquad \mathcal L_j=L(1,\Ad\phi_j), \] and \[ h_{T,R}^{(n)}(\alpha)=\bigl|p_{T,R}^{n,\#}(\alpha)\bigr|^2 \Big/ \prod_{j\ne k}\Gamma\Bigl(\tfrac{1+\alpha_j-\alpha_k}2\Bigr), \] then the conjectural limit is \[ \frac{\sum_j\lambda_j(\ell)\,\overline{\lambda_j(m)}\, \tfrac{h_{T,R}^{(n)}(\alpha^{(j)})}{\mathcal L_j}} {\sum_j \tfrac{h_{T,R}^{(n)}(\alpha^{(j)})}{\mathcal L_j}} \;\longrightarrow\; \begin{cases} 1,&\ell=m,\\ 0,&\ell\ne m, \end{cases} \qquad (T\to\infty), \] with a power-saving error term once the test function is chosen suitably [2212.14534]. For cuspidal automorphic representations over a number field $F$, [2507.20653] states Selberg’s orthogonality conjecture in prime-average form. If $\pi\in\mathfrak F_n$ and $\pi'\in\mathfrak F_{n'}$, then as $x\to\infty$, \[ \sum_{N p \le x} a_\pi(p)\,a_{\pi'}(p)/N p = \delta_{\pi,\pi'}\,\log\log x + O_{\pi,\pi'}(1). \] This is the formulation that arises from Rankin–Selberg coefficients and Hypothesis H of Rudnick and Sarnak [2507.20653]. ## 2. Classical and low-rank cases The case $N=2$ is the classical Selberg orthogonality obtained through the Kuznetsov formula. Here $\phi_j$ runs over $\SL(2,\Z)$ Maass cusp forms, $A_j(n)=a_j(n)$ are Hecke–Fourier coefficients, and \[ L(1,\phi_j,\mathrm{Ad})=L(1,\mathrm{sym}^2\phi_j). \] Bruggeman proved that for any $m,n\ge 1$, \[ \lim_{T\to\infty} \frac{\sum_{t_j\le T}a_j(m)\,\overline{a_j(n)}\,h_T(t_j)\,/\,L(1,\mathrm{sym}^2\phi_j)} {\sum_{t_j\le T}h_T(t_j)\,/\,L(1,\mathrm{sym}^2\phi_j)} = \delta_{m,n}, \] which [1303.0889] describes as the classical Selberg, or Kuznetsov–Petersson, orthogonality. For $N=3$, Goldfeld–Kontorovich and independently Blomer established the full weighted orthogonality with power-saving error. If $A_j(m_1,m_2)$ are normalized $\SL(3,\Z)$ Fourier coefficients, $P=m_1m_2n_1n_2$, and $\theta\le 7/64$ is any admissible bound towards $\GL(2)$-Ramanujan, then \[ \frac{\sum_j A_j(m_1,m_2)\,\overline{A_j(n_1,n_2)}\;h_T(v_j)\,/\,L(1,\phi_j,\mathrm{Ad})} {\sum_j h_T(v_j)\,/\,L(1,\phi_j,\mathrm{Ad})} = \delta_{(m_1,m_2),(n_1,n_2)} +O\Bigl((T^2P^{1/2}+T^3P^{\theta}+P^{5/3})\,(T\,P)^{\varepsilon}\Bigr) \] as $T\to\infty$ [1303.0889]. This provides the low-rank model for Zhou’s higher-rank conjecture and for the vertical equidistribution results derived from it. The low-rank record is broadened in [2212.14534], which establishes an explicit asymptotic orthogonality relation with power-saving error for $\GL(n,\R)$ when $2\le n\le 5$. In particular, the case $n=5$ is identified there as a new result. For $n>5$, the same paper obtains a conditional theorem under two auxiliary conjectures. ## 3. Kuznetsov-trace-formula framework on $\GL(n,\mathbb R)$ The analysis in [2212.14534] is organized around a higher-rank Kuznetsov trace formula with a test function $p_{T,R}^{(n)}$ whose transform localizes spectral parameters up to size $T$. On the spectral side, one encounters \[ \sum_{\phi\;\rm cusp}\lambda_\phi(\ell)\overline{\lambda_\phi(m)} \,\frac{\bigl|p_{T,R}^{n,\#}(\alpha_\phi)\bigr|^2}{\<\phi,\phi\>} \;+\; \sum_{\Eis}A_{\Eis}(\ell,s)\,\overline{A_{\Eis}(m,s)}\, \bigl|p_{T,R}^{n,\#}(\alpha(s))\bigr|^2\,ds, \] while on the geometric side the identity term $\mathcal M$ contributes the diagonal main term and the remaining Weyl elements contribute Kloosterman-sum integrals $\mathcal I_w$ [2212.14534]. The latter are shown to satisfy \[ \mathcal I_w=\O(T^{\kappa_1-1+\varepsilon}(\ell m)^A), \] where \[ \kappa_1=R\bigl(\tbinom{2n}{n}-2^n\bigr)+n. \] The resulting orthogonality theorem for $2\le n\le 5$ has the form \[ \sum_j\lambda_j(\ell)\,\overline{\lambda_j(m)}\, \frac{h_{T,R}^{(n)}(\alpha^{(j)})}{\mathcal L_j} = \delta_{\ell,m}\sum_{i=1}^{n-1} \mathfrak c_i\,T^{\kappa_i} +\O\bigl((\ell m)^{A}\,T^{\kappa_1-1+\varepsilon}\bigr), \] where the $\kappa_i$ form a descending sequence of exponents of size $T^{\kappa_1}$ [2212.14534]. For $n>5$ the same asymptotic is conditional on two hypotheses: a lower bound for Rankin–Selberg $L(1+it,\phi\times\phi')$ on $\Re s=1$, and an Ishii–Stade type recurrence for Mellin transforms of $\GL(n)$ Whittaker functions. Several structural identities enter the proof. The first Fourier coefficient of a Maass cusp form satisfies \[ \<\phi,\phi\>=c_n\,L(1,\Ad\phi)\prod_{j\ne k}\Gamma\Bigl(\tfrac{1+\alpha_j-\alpha_k}2\Bigr), \] Eisenstein Fourier coefficients factor into products of Rankin–Selberg $L$-values and archimedean gamma-factors, and the Whittaker Mellin transform satisfies explicit Stade-type recurrences [2212.14534]. The paper also states that temperedness at the real place is assumed, but that one may remove or relax this assumption with a minor weakening in the error term. ## 4. Higher-rank conjecture on $\PGL(N)$ and vertical Sato–Tate For $N\ge 4$, Zhou formulates the higher-rank Selberg orthogonality conjecture on $\PGL(N)$ exactly in the form of the weighted Fourier-coefficient limit described above, with $h_T$ sharply cutting to $\|v\|\le T$ and weight $1/L(1,\phi_j,\mathrm{Ad})$ [1303.0889]. The paper isolates three obstructions to proving this directly in higher rank: the lack of an explicit Kuznetsov or relative trace formula rich enough to isolate individual Fourier modes, the more complicated root-system combinatorics and the presence of many Whittaker integrals, and the necessity of controlling continuous-spectrum and Eisenstein contributions. The principal consequence is a weighted vertical equidistribution theorem for Satake parameters at a fixed finite prime $p$. If \[ X_j(p)\in T/W \] is the Satake parameter of $\phi_j$ at $p$, where $T\simeq (\C^*)^N\cap\det=1$ and $W=S_N$, then assuming the orthogonality conjecture for $N\ge 4$ one has \[ \lim_{T\to\infty} \frac{\displaystyle \sum_j f\bigl(X_j(p)\bigr)\;h_T\bigl(v(\phi_j)\bigr)\,/\,L\bigl(1,\phi_j,\mathrm{Ad}\bigr)} {\displaystyle \sum_j h_T\bigl(v(\phi_j)\bigr)\,/\,L\bigl(1,\phi_j,\mathrm{Ad}\bigr)} = \int_{T_0/W} f(x)\,dx \] for every continuous $f:T/W\to\C$, where $T_0=SU(N)\cap T$ and $dx$ is the normalized Sato–Tate measure on $T_0/W$ [1303.0889]. In the notation of [1303.0889], identifying \[ T_0\simeq\{(e^{i\theta_1},\dots,e^{i\theta_N}):\sum\theta_j\equiv0\pmod{2\pi}\}, \] the Weyl-pushforward of Haar measure on $SU(N)$ gives the density \[ dx \;=\; \frac{1}{N!\,(2\pi)^{N-1}\, \prod_{1\le i0. \] In particular, the paper records exponential-in-$\sqrt{\log x}$ decay in the diagonal error and arbitrary power-saving in $\log x$ off the diagonal [2507.20653]. The proof is built on three analytic inputs. The first is an effective bound on Euler products of Rankin–Selberg coefficients: \[ \prod_p \Bigl(1 + \sum_{k=2}^\infty \lambda_{\pi\times\widetilde\pi}(p^k)/N p^{k\sigma}\Bigr) \ll C(\pi)^\varepsilon \qquad \text{for all }\sigma \ge 1 - 1/(n^2+1) + \varepsilon. \] The second is a power sieve over number fields: \[ \sum_{n} f(n^k) \;\ll\; P^{-2} D_F^{O(k^{[F:\Q]+1})} \sum_{n} f(n)\,\Bigl(\bigl|\sum_{p\in\mathcal P,\;N p\le P} \chi_p(n)\,\log N p\bigr|^2 + \log^2(2+N n)\Bigr). \] The third is an iterative argument à la Iwaniec, which bootstraps a weak bound at $\sigma_0=1-1/(n^2+1)+\varepsilon$ to a $C(\pi)^\varepsilon$ bound for any $\sigma>\sigma_0$ [2507.20653]. A further ingredient is the factorization \[ L(s,\pi\otimes\pi')=L(s,\pi\times\pi')\,H(s,\pi,\pi'), \qquad H(s,\pi,\pi')\ll (C(\pi)C(\pi'))^\varepsilon \] for $\sigma>1-1/(n^2+1)$ [2507.20653]. The paper emphasizes that this framework is unconditional in all ranks and bypasses the functoriality barrier that had previously limited such arguments to $n\le 4$. ## 6. Barriers, comparisons, and mathematical significance The higher-rank spectral problem remains shaped by trace-formula limitations. In [1303.0889], the main barrier for $N\ge 4$ is the absence of a workable higher-rank Kuznetsov trace formula that isolates individual Fourier modes with sufficiently sharp test functions. In [2212.14534], the general $\GL(n,\R)$ theorem for $n>5$ is conditional on a lower-bound conjecture for Rankin–Selberg $L$-functions on $\Re s=1$ and a recurrence conjecture for Mellin transforms of $\GL(n)$ Whittaker functions. These are not competing formulations but distinct higher-rank fronts of the same orthogonality problem. The literature summarized in [2507.20653] places the recent advances in context. Rudnick–Sarnak proved Selberg orthogonality only for $n\le 3$ over $\Q$, and for $n=4$ under Hypothesis H or exterior-square functoriality. Liu–Ye obtained uniform $O(1)$ off-diagonal and $O(e^{-c\sqrt{\log x}})$ diagonal estimates under the full Ramanujan conjecture and self-contragredience. Liu–Wang–Ye removed Ramanujan but retained Hypothesis H and self-contragredience; Avdispahić–Smajlović removed self-contragredience but still assumed Hypothesis H. Wu–Ye and Humphries–Thorner studied prime number theorems for fixed $\pi$ up to $n=4$, but not full orthogonality in all ranks [2507.20653]. The conjecture has several direct consequences. After normalization by the harmonic weights $1/L(1,\Ad\phi)$, [2212.14534] states that Hecke eigenvalues $\lambda_\phi(p)$ become equidistributed with respect to the correct Sato–Tate measure, independently of $p$. Zhou’s paper shows that, in rank $2$, vertical equidistribution for Hecke eigenvalues with weight $1/L(1,\Ad)$ is essentially equivalent to orthogonality of Fourier coefficients, and suggests that sufficiently strong vertical equidistribution at all primes might force orthogonality through multi-prime moment methods [1303.0889]. The number-field theorem of [2507.20653] further yields unconditional GUE statistics for automorphic $L$-function zeros, the first effective polynomial bound for strong multiplicity one for coefficients, prime number theorems for Rankin–Selberg $L$-functions in long and short intervals, and Hoheisel-type theorems. Taken together, these results show that Selberg orthogonality is both a structural statement about automorphic spectra and a central analytic input for the modern theory of automorphic $L$-functions. Source: https://www.emergentmind.com/topics/selberg-orthogonality-conjecture