Selberg Orthogonality Conjecture

• Selberg Orthogonality Conjecture is a statement on the asymptotic orthogonality of automorphic Fourier coefficients and Hecke eigenvalues, connecting rank-one and higher-rank trace formulas.
• It uses harmonic weights and spectral localization to derive effective error estimates and equidistribution results for Maass forms on groups like GL(n) and PGL(N).
• The conjecture underpins advances in automorphic L-functions by providing structural insights that lead to strong multiplicity one, prime number theorem refinements, and GUE statistics.

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 (Zhou, 2013, Goldfeld et al., 2022, Jiang, 28 Jul 2025).

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

$\PGL(N)$0

If $\PGL(N)$1 is the spectral parameter and $\PGL(N)$2 is a smooth cut-off which for large $\PGL(N)$3 essentially restricts to $\PGL(N)$4, Zhou defines the harmonic weight

$\PGL(N)$5

The conjectured orthogonality relation is

$\PGL(N)$6

where $\PGL(N)$7 and $\PGL(N)$8 are tuples of positive integers, and $\PGL(N)$9 exactly when $1/L(1,\mathrm{Ad})$0 for all $1/L(1,\mathrm{Ad})$1 (Zhou, 2013). 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 (Goldfeld et al., 2022) for Hecke eigenvalues of Maass cusp forms on $1/L(1,\mathrm{Ad})$2. If

$1/L(1,\mathrm{Ad})$3

and

$1/L(1,\mathrm{Ad})$4

then the conjectural limit is

$1/L(1,\mathrm{Ad})$5

with a power-saving error term once the test function is chosen suitably (Goldfeld et al., 2022).

For cuspidal automorphic representations over a number field $1/L(1,\mathrm{Ad})$6, (Jiang, 28 Jul 2025) states Selberg’s orthogonality conjecture in prime-average form. If $1/L(1,\mathrm{Ad})$7 and $1/L(1,\mathrm{Ad})$8, then as $1/L(1,\mathrm{Ad})$9,

$\log\log x$0

This is the formulation that arises from Rankin–Selberg coefficients and Hypothesis H of Rudnick and Sarnak (Jiang, 28 Jul 2025).

2. Classical and low-rank cases

The case $\log\log x$1 is the classical Selberg orthogonality obtained through the Kuznetsov formula. Here $\log\log x$2 runs over $\log\log x$3 Maass cusp forms, $\log\log x$4 are Hecke–Fourier coefficients, and

$\log\log x$5

Bruggeman proved that for any $\log\log x$6,

$\log\log x$7

which (Zhou, 2013) describes as the classical Selberg, or Kuznetsov–Petersson, orthogonality.

For $\log\log x$8, Goldfeld–Kontorovich and independently Blomer established the full weighted orthogonality with power-saving error. If $\log\log x$9 are normalized $\PGL(N)$0 Fourier coefficients, $\PGL(N)$1, and $\PGL(N)$2 is any admissible bound towards $\PGL(N)$3-Ramanujan, then

$\PGL(N)$4

as $\PGL(N)$5 (Zhou, 2013). 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 (Goldfeld et al., 2022), which establishes an explicit asymptotic orthogonality relation with power-saving error for $\PGL(N)$6 when $\PGL(N)$7. In particular, the case $\PGL(N)$8 is identified there as a new result. For $\PGL(N)$9, the same paper obtains a conditional theorem under two auxiliary conjectures.

3. Kuznetsov-trace-formula framework on $\{\phi_j\}$0

The analysis in (Goldfeld et al., 2022) is organized around a higher-rank Kuznetsov trace formula with a test function $\{\phi_j\}$1 whose transform localizes spectral parameters up to size $\{\phi_j\}$2. On the spectral side, one encounters

$\{\phi_j\}$3

while on the geometric side the identity term $\{\phi_j\}$4 contributes the diagonal main term and the remaining Weyl elements contribute Kloosterman-sum integrals $\{\phi_j\}$5 (Goldfeld et al., 2022). The latter are shown to satisfy

$\{\phi_j\}$6

where

$\{\phi_j\}$7

The resulting orthogonality theorem for $\{\phi_j\}$8 has the form

$\{\phi_j\}$9

where the $$L^2_\mathrm{cusp}\bigl(SL(N,\Z)\backslash GL(N,\R)/O(N)\bigr),$$0 form a descending sequence of exponents of size $$L^2_\mathrm{cusp}\bigl(SL(N,\Z)\backslash GL(N,\R)/O(N)\bigr),$$1 (Goldfeld et al., 2022). For $$L^2_\mathrm{cusp}\bigl(SL(N,\Z)\backslash GL(N,\R)/O(N)\bigr),$$2 the same asymptotic is conditional on two hypotheses: a lower bound for Rankin–Selberg $$L^2_\mathrm{cusp}\bigl(SL(N,\Z)\backslash GL(N,\R)/O(N)\bigr),$$3 on $$L^2_\mathrm{cusp}\bigl(SL(N,\Z)\backslash GL(N,\R)/O(N)\bigr),$$4, and an Ishii–Stade type recurrence for Mellin transforms of $$L^2_\mathrm{cusp}\bigl(SL(N,\Z)\backslash GL(N,\R)/O(N)\bigr),$$5 Whittaker functions.

Several structural identities enter the proof. The first Fourier coefficient of a Maass cusp form satisfies

$$L^2_\mathrm{cusp}\bigl(SL(N,\Z)\backslash GL(N,\R)/O(N)\bigr),$$6

Eisenstein Fourier coefficients factor into products of Rankin–Selberg $$L^2_\mathrm{cusp}\bigl(SL(N,\Z)\backslash GL(N,\R)/O(N)\bigr),$$7-values and archimedean gamma-factors, and the Whittaker Mellin transform satisfies explicit Stade-type recurrences (Goldfeld et al., 2022). 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 $$L^2_\mathrm{cusp}\bigl(SL(N,\Z)\backslash GL(N,\R)/O(N)\bigr),$$8 and vertical Sato–Tate

For $$L^2_\mathrm{cusp}\bigl(SL(N,\Z)\backslash GL(N,\R)/O(N)\bigr),$$9, Zhou formulates the higher-rank Selberg orthogonality conjecture on $A_j(m_1,\dots,m_{N-1})$0 exactly in the form of the weighted Fourier-coefficient limit described above, with $A_j(m_1,\dots,m_{N-1})$1 sharply cutting to $A_j(m_1,\dots,m_{N-1})$2 and weight $A_j(m_1,\dots,m_{N-1})$3 (Zhou, 2013). 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 $A_j(m_1,\dots,m_{N-1})$4. If

$A_j(m_1,\dots,m_{N-1})$5

is the Satake parameter of $A_j(m_1,\dots,m_{N-1})$6 at $A_j(m_1,\dots,m_{N-1})$7, where $A_j(m_1,\dots,m_{N-1})$8 and $A_j(m_1,\dots,m_{N-1})$9, then assuming the orthogonality conjecture for $(m_1,\dots,m_{N-1})$0 one has

$(m_1,\dots,m_{N-1})$1

for every continuous $(m_1,\dots,m_{N-1})$2, where $(m_1,\dots,m_{N-1})$3 and $(m_1,\dots,m_{N-1})$4 is the normalized Sato–Tate measure on $(m_1,\dots,m_{N-1})$5 (Zhou, 2013).

In the notation of (Zhou, 2013), identifying

$(m_1,\dots,m_{N-1})$6

the Weyl-pushforward of Haar measure on $(m_1,\dots,m_{N-1})$7 gives the density

$(m_1,\dots,m_{N-1})$8

The paper states that, by the Weyl integration formula, this is the unique probability measure invariant under the Weyl group.

For $(m_1,\dots,m_{N-1})$9, Zhou proves an explicit rate of convergence. If $\phi_j$0 is any Laurent monomial in the two fundamental characters $\phi_j$1 of $\phi_j$2, then

$\phi_j$3

with $\phi_j$4 as above (Zhou, 2013). The mechanism is representation-theoretic: via the Casselman–Shalika formula, moments of Fourier coefficients are converted into sums of characters $\phi_j$5 against the harmonic weight, and then Peter–Weyl yields equidistribution.

5. Effective prime-average orthogonality over number fields

The 2025 paper (Jiang, 28 Jul 2025) proves Hypothesis H in full generality for $\phi_j$6 over any number field and uses it to establish Selberg orthogonality with explicit error terms. For $\phi_j$7 and $\phi_j$8, the theorem distinguishes three cases.

On the diagonal, when $\phi_j$9,

$\PGL(N)$00

For a twist by a unitary character, when $\PGL(N)$01 with $\PGL(N)$02 real,

$\PGL(N)$03

Off the diagonal, when $\PGL(N)$04 for any $\PGL(N)$05,

$\PGL(N)$06

In particular, the paper records exponential-in-$\PGL(N)$07 decay in the diagonal error and arbitrary power-saving in $\PGL(N)$08 off the diagonal (Jiang, 28 Jul 2025).

The proof is built on three analytic inputs. The first is an effective bound on Euler products of Rankin–Selberg coefficients: $\PGL(N)$09 The second is a power sieve over number fields: $\PGL(N)$10 The third is an iterative argument à la Iwaniec, which bootstraps a weak bound at $\PGL(N)$11 to a $\PGL(N)$12 bound for any $\PGL(N)$13 (Jiang, 28 Jul 2025).

A further ingredient is the factorization

$\PGL(N)$14

for $\PGL(N)$15 (Jiang, 28 Jul 2025). The paper emphasizes that this framework is unconditional in all ranks and bypasses the functoriality barrier that had previously limited such arguments to $\PGL(N)$16.

6. Barriers, comparisons, and mathematical significance

The higher-rank spectral problem remains shaped by trace-formula limitations. In (Zhou, 2013), the main barrier for $\PGL(N)$17 is the absence of a workable higher-rank Kuznetsov trace formula that isolates individual Fourier modes with sufficiently sharp test functions. In (Goldfeld et al., 2022), the general $\PGL(N)$18 theorem for $\PGL(N)$19 is conditional on a lower-bound conjecture for Rankin–Selberg $\PGL(N)$20-functions on $\PGL(N)$21 and a recurrence conjecture for Mellin transforms of $\PGL(N)$22 Whittaker functions. These are not competing formulations but distinct higher-rank fronts of the same orthogonality problem.

The literature summarized in (Jiang, 28 Jul 2025) places the recent advances in context. Rudnick–Sarnak proved Selberg orthogonality only for $\PGL(N)$23 over $\PGL(N)$24, and for $\PGL(N)$25 under Hypothesis H or exterior-square functoriality. Liu–Ye obtained uniform $\PGL(N)$26 off-diagonal and $\PGL(N)$27 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 $\PGL(N)$28 up to $\PGL(N)$29, but not full orthogonality in all ranks (Jiang, 28 Jul 2025).

The conjecture has several direct consequences. After normalization by the harmonic weights $\PGL(N)$30, (Goldfeld et al., 2022) states that Hecke eigenvalues $\PGL(N)$31 become equidistributed with respect to the correct Sato–Tate measure, independently of $\PGL(N)$32. Zhou’s paper shows that, in rank $\PGL(N)$33, vertical equidistribution for Hecke eigenvalues with weight $\PGL(N)$34 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 (Zhou, 2013). The number-field theorem of (Jiang, 28 Jul 2025) further yields unconditional GUE statistics for automorphic $\PGL(N)$35-function zeros, the first effective polynomial bound for strong multiplicity one for coefficients, prime number theorems for Rankin–Selberg $\PGL(N)$36-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 $\PGL(N)$37-functions.