Probing automorphic induction through zero correlations and secondary arithmetic terms

Yangbo Ye

REVIEW

Probing automorphic induction through zero correlations and secondary arithmetic terms

Yangbo Ye

Number Theory · Submitted 1 Nov 2026 · 27 references

Pith's verdict: Serious, honest paper: the probe statistic and the mod-13 example are real contributions, but the central subexponential theorem is conditional on an unreviewed preprint (Jiang, arXiv:2507.20653v1) that the paper itself has to patch.

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arXiv ID2611.02058 (version 1)

Pith IDV3GAKB6A, Pith's permanent record of this paper and its review

CitationYangbo Ye. "Probing automorphic induction through zero correlations and secondary arithmetic terms." arXiv:2611.02058 (2026). https://pith.science/paper/V3GAKB6A

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Claim paper

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Editor's note

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The one thing to know: this paper deserves a serious referee, and the referee's main job is to check one external input. The internal argument I sampled is coherent and careful, and the paper is unusually honest about its own limits.

What is genuinely new is the probe idea. For a product F and a fixed cuspidal P, the exact identity (1/2)(R_{FP,FP} − R_{F,F} − R_{P,P}) = R_{F,P} turns mixed zero correlation into a constituent test. With a Gevrey-order-two amplitude supported away from zero, the smooth-density and finite-part terms vanish, and the leading constant is exactly the multiplicity of π in the assumed factorization, with no conductor or gamma subtraction. That is not a restatement of anything in the cited literature. The secondary-term hierarchy — mixed finite parts C_{F,G}, connected prime-power cumulants, the first connected triple contribution at (log T)^{-3}, the polarization identities — is worked out in detail.

The best standalone piece is Section 12.1: two products of four primitive characters mod 13 with identical gamma factors, conductor, and complete self-pair arithmetic series, separated by Γ_3, with an exact difference formula and an interval-arithmetic certificate. That part is self-contained and checkable, and needs no Jiang input.

The soft spot is exactly where the reader and stress-test put it, and the CONDITIONAL verdict is right. The subexponential error in the constituent probe runs through Theorem 6.2, whose local expansion of A_{F,G}(1+z) needs the quantitative prime-power tail (H^δ_{F,G}) proved in Proposition 6.1 from Jiang's rational-field fixed-data theorem — an unreviewed preprint, one the paper itself flags as needing a fix in a displayed bound. This is not a manufactured flaw; it is load-bearing. The dependence also reaches further than the subexponential rate: the same source supplies the prime-power convergence behind the qualitative secondary term, so even the o(1) detection statement at general rank sits on it. The Liu–Ye strengthening additionally leans on a submitted corrigendum, though the Gevrey route makes that optional.

Everything on top of that input that I checked is internally consistent: the probe identity, the plateau lemma, the balanced-packet reduction, and the explicit refusal to transfer the pair remainder to higher levels. The limitation statements are complete rather than cosmetic.

Who it is for: people working on zero statistics for automorphic L-functions and on extracting arithmetic data from lower-order terms. Send it to peer review, and tell the referee to scrutinize Section 6.1's deduction from Jiang and the preprint's status.

What would settle it

The decisive check is the imported tail itself: for a fixed unitary cuspidal pair over $\mathbb{Q}$, estimate $\sum_{p^k > Y,\, k \ge 2} (k-1)(\log p)^2 |a_\sigma(p^k)a_\tau(p^k)| p^{-k}$ at large $Y$, since any failure to decay like $O(Y^{-\delta})$ with $\delta > 0$ would break the meromorphic input behind the subexponential error. On the side that needs no unproved input, the level-three claim is concrete: for the two products of primitive Dirichlet characters modulo $13$, the paper's formula gives $\Gamma_3(F_A) - \Gamma_3(F_B) = 24\sqrt{3}\sum_p \varepsilon(p)(\log p)^3 R(p) \in (-1.503, -1.228)$, a range an independent reader can certify by direct summation with a rigorous tail bound, and landing outside that interval would refute the claimed gain of level three over all pair-level self data.

Referee report

Significance

If the external arithmetic inputs are accepted, this is a substantial and useful contribution: it gives explicit secondary arithmetic terms in raw correlations, a clean detection-and-multiplicity criterion for a prescribed cuspidal constituent, and new higher-level connected invariants. The manuscript has real strengths: the exact identity (7.1), the Gevrey localization construction in Lemma 5.2, the purely local argument in Theorem 6.2, the balanced-packet reduction in Proposition 8.1, the explicit distinct-index sieve, and the interval-arithmetic certificate in Section 12.1. It is also honest about its scope: no uniformity in twists, no interchange of n and T limits, no general reconstruction theorem, and explicit non-injectivity of the self hierarchy. The main obstacle is external: the quantitative prime-power tail and the subexponential pair refinement are imported from an unreviewed preprint, with a second subexponential input coming from a submitted corrigendum.

Major objections (3)

[§6.1, Prop. 6.1; Thm. 6.2; Prop. 7.1] The central probe theorem rests on an unreviewed preprint in an essential way. Proposition 6.1 derives the quantitative tail (H^delta_{F,G}) from (6.1), which is the rational-field case of Jiang's Theorem 1.3 in arXiv:2507.20653v1; that tail is then used to make H_{F,G}(1+z) holomorphic near z=0, giving the expansion (6.7), Theorem 6.2, and the O(e^{-a sqrt(log T)}) error in Proposition 7.1. The manuscript itself notes in §6.1 that a displayed bound in the preprint must be corrected, namely that it 'must retain its dependence on s-1', so the cited source is not stable. Since the main claim of the paper should not depend on an unpublished and self-flagged source without explicit qualification, please either supply a self-contained proof of the needed rational-field fixed-data estimate or state Theorem 6.2 and Proposition 7.1 as conditional on Jiang's Theorem 1.3, and separate clearly which results are unconditional.
[§9.1, Thm. 10.1] The same external input controls the higher-level expansion. The convergence of the resonance cumulants Gamma_B in §9.1 is proved from (J+), which is exactly the fixed-data consequence (6.1) of Jiang's preprint; Theorem 10.1 and the fourth-level formula (11.1) therefore inherit the conditional status of that input. The degree-one example in §12.1 may be elementary, but as written it is interpreted through Theorem 10.1; please either provide the elementary justification for the degree-one case or state explicitly how that case bypasses the general convergence machinery. This broadens the external-support concern beyond the pair probe, although it is not an internal contradiction.
[§6.2, Thm. 2.2, Cor. 2.4-2.5] The stronger subexponential prime number theorem (PNT_{F,G}) is quoted from Liu-Ye [18] together with a 'corrected proof and ramified local treatment in the submitted corrigendum [26]'. Since [26] is a submitted, not yet published, corrigendum, the statements that use (PNT_{F,G}) — namely (2.17) and the self-contragredient cases of Corollaries 2.4 and 2.5 — should be labeled conditional on that corrigendum, or the needed correction should be included in the present paper. This does not affect Proposition 7.1, which uses the local Gevrey argument, but it affects the scope of the plateau-refinement claims.

Minor objections (4)

[§6.1] The sentence 'We cite the version reviewed here' is misleading for an arXiv preprint; please replace it with a neutral description such as 'the version currently posted' and make explicit that the preprint has not yet appeared in a peer-reviewed venue.
[§2, Cor. 2.4] The phrase 'No self-contragredience, additional Hypothesis H, or temperedness assumption is required' is easy to misread as 'no external arithmetic input is required', since Corollary 2.4 uses Proposition 6.1. Please reformulate, for example as 'no Hypothesis H beyond the quantitative tail supplied by Proposition 6.1'.
[§11.3, Eq. (11.2)] In the distinct-index sieve, the symbol F is used both for the fixed product and for a partition of [n]; this overloading makes (11.2) harder to read. Suggest renaming the partition variable, e.g. Q or Pi.
[§12.1] The numerical certificate is welcome, but the sentence 'There are 26169 primes in the four relevant residue classes in this finite sum' should specify the exact residue classes and the sieve procedure, so that the count is reproducible without reverse-engineering the periodicity condition.

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The paper

The paper claims

Adjoining a probe L-function and differencing three raw zero correlations detects a specified cuspidal factor of an L-function product — and returns its exact multiplicity.

The reading

Zero correlations of L-function zeros are coarse statistics whose leading self terms record only aggregated data such as the total degree and the sum of squared multiplicities. This paper argues that adjoining a fixed probe L-function and differencing three raw correlations turns them into a precise detector: a Fourier test supported away from zero makes the leading term of the isolated mixed correlation equal to the multiplicity of the probe inside the assumed cuspidal factorization of the product, with subexponential error. The same expansion machinery then computes the full inverse-logarithmic series of raw mixed $n$-level correlations, exposing connected prime-power invariants whose first new contribution appears at order $(\log T)^{-3}$, and a worked example shows two products sharing all gamma factors and all self-pair arithmetic data that differ in that cubic constant. If the central claim is right, raw zero correlations become a working probe of automorphic induction structure rather than a source of coarse aggregates alone.

What carries the argument

The carrying object is the raw mixed correlation $R_{F,G}(T)$ — a height average, with a common normalization, over pairs of ordinates of the completed $L$-functions $F$ and $G$ — evaluated through the explicit formula in the strict small-support range $cb<1$. Three devices do the work. The identity $\frac12(R_{FP,FP} - R_{F,F} - R_{P,P}) = R_{F,P}$ is exact because zero multisets add under products, so probe detection reduces to one mixed pair correlation; the paper stresses that no product may be rescaled by its own degree before differencing. The arithmetic finite part $C_{F,G} = \lim_{s\to 1^+}(A_{F,G}(s) - r_{F,G}(s-1)^{-2})$ organizes the coefficient series $A_{F,G}(s) = \sum_{p,k} (\log p)^2 w_{F,G}(p^k)p^{-ks}$; its local Laurent expansion near $s=1$, secured by the quantitative prime-power tail of Proposition 6.1, lets a Gevrey-order-two test function (one with $|\phi^{(j)}|_\infty \le C_0 A_0^j (j!)^2$, so its Fourier transform decays like $e^{-a\sqrt{|t|}}$) replace the global prime number theorem by a purely local argument with error $e^{-a\sqrt{\log T}}$. At levels three and higher, a moment–cumulant expansion over prime-power resonances — relations $\sum_j \varepsilon_j k_j = 0$ such as $p^a p^b = p^{a+b}$ — defines connected invariants $\Gamma_B$ whose first contribution to a normalized correlation appears at order $(\log T)^{-3}$.

Extended reading notes

Core claim

The central claim is Proposition 7.1: for $F = L(s,\Pi)$ assumed to factor as in (1.1) into inequivalent unitary cuspidal representations over $\mathbb{Q}$, and for a fixed unitary cuspidal probe $P = L(s,\pi)$, any nonzero, nonnegative, even Gevrey-order-two Fourier amplitude $\phi^*$ supported away from zero gives $$\frac{$R^{{\phi^*}}$_{FP,FP}(T) - $R^{{\phi^*}}$_{F,F}(T) - $R^{{\phi^*}}$_{P,P}(T)}{2J_{\phi^*}} = r_\Pi(\pi) + O($e^{{-a\sqrt{\log T}}$}),$$ where $r_\Pi(\pi)$ is the multiplicity of $\pi$ in the factorization and is zero when $\pi$ is absent. The identity behind the detector is exact at finite height — half the alternating sum of the three raw correlations equals the mixed correlation $R_{F,P}(T)$ — and the choice $\phi^*(0) = 0$ eliminates every smooth-density and finite-part term, leaving the overlap coefficient $r_{F,P}J_{\phi^*}$ plus a subexponential remainder produced by a local meromorphic argument near $s=1$. The paper further establishes a full expansion of raw mixed $n$-level correlations at every fixed level and inverse-logarithmic order, whose coefficients are archimedean terms, Laurent coefficients of mixed Rankin–Selberg series, and moments of connected prime-power cumulants; the first genuinely three-coordinate invariant $\Gamma_3$ enters at order $(\log T)^{-3}$ through relations such as $p^a p^b = p^{a+b}$. An explicit example with products of primitive Dirichlet characters modulo $13$ shows two $L$-functions with identical gamma factors and identical complete self-pair arithmetic series but different $\Gamma_3$ values, proving that level-three data carry arithmetic information invisible to every self-pair statistic.

Load-bearing premise

The detector presumes the factorization (1.1) exists, and it imports rather than proves the quantitative prime-power tail of Proposition 6.1 — a power-saving bound on $\sum_{p^k > Y,\, k\ge 2} (k-1)(\log p)^2 |w_{F,G}(p^k)| p^{-k}$ quoted from an unreviewed preprint; if that tail fails for some fixed product pair, the meromorphic expansion near $s=1$ and the subexponential error in Proposition 7.1 collapse.

Editorial extensions

If this is right

A positive probe response identifies a constituent of the specified degree and returns its multiplicity; applying the probe to a finite candidate list and totalling the detected degrees certifies when the list is exhaustive, as a stopping certificate rather than an effective algorithm.
Because the chosen test vanishes near zero, the detector needs no subtraction of conductor or gamma-factor contributions: the alternating sum of three raw correlations removes all smooth-density terms by itself.
Level-three mixed correlations are strictly richer than the entire self-pair hierarchy: two products of Dirichlet characters modulo 13 share all gamma factors and all self-pair arithmetic coefficients yet differ in the cubic constant $\Gamma_3$.
Varying integer powers of the probe (replacing $F$ by $FP^z$) and interpolating in $z$ separates mixed higher invariants such as $\Gamma(F,F,P)$ and $\Gamma(F,P,P)$ from each other.
For arbitrary smooth plateau tests the pair remainder is smaller than every fixed inverse power of $\log T$, with no self-contragredience assumption, and for Gevrey-order-two tests the remainder is subexponential for every fixed product pair.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The probe identity holds at finite height for any zero statistic, so the same product-minus-factors background subtraction could in principle be applied to low-lying zeros, distinct-index correlations, or other statistics whenever a probe's zero set can be superimposed on the target's; the paper develops only the height-averaged raw case.
The Gevrey refinement is purely local near $s=1$, which suggests the subexponential pair formula can be transplanted to any $L$-function setting with a known meromorphic continuation of the coefficient series in a fixed neighborhood of $1$, bypassing quantitative prime number theorems altogether.
The main probe error is asymptotic, not effective; obtaining explicit dependence on the conductor, the probe, and the test functions would turn the detector into a computational tool for recovering cuspidal decompositions, a step the paper states it does not take.
Because all self-correlation data collapse for characters of the same conductor, fine constituent identification must come from mixed probe statistics; the polarization formulas extract such mixed invariants, but whether any finite probe family is injective for the factorization is left open.

Original abstract

Zero correlations offer a way to test the cuspidal structure of an automorphic L-function, but their leading self statistics retain only coarse degree and multiplicity data. We therefore adjoin a specified cuspidal L-function as a probe and study the resulting raw mixed correlations. Assuming a factorization into cuspidal L-functions over Q, we show that a mixed pair statistic detects the probe's occurrence and recovers its multiplicity. A suitable Gevrey-order-two test eliminates the smooth-density terms and gives a subexponential error. For arbitrary smooth plateau tests, the prime number theorem of Harcos and Thorner gives an error smaller than every fixed inverse power of log T, without a self-contragredience assumption. The stronger prime number theorem of Liu and Ye gives a subexponential remainder under its stated structural assumptions, with the possible norm-twist terms made explicit. To obtain arithmetic information beyond this overlap, we compute the correlation expansion at every fixed level and inverse-logarithmic order within the original strict small-support range. We treat the pair and higher correlations separately, using Jiang's rational-field fixed-data estimates to justify the required prime-power convergence. The pair calculation identifies the mixed arithmetic finite parts and gives detailed error estimates, including the contribution of norm twists. The higher-level calculation retains connected prime-power invariants, whose first cubic contribution appears at order (log T)^{-3}. Explicit third- and fourth-level formulas and examples explain the arithmetic content of these coefficients and the limitations of self-correlations. Varying powers of the probe separates their mixed components. These formulas, together with the distinct-index sieve, provide a collection of arithmetic probe responses toward the further problem of recovering automorphic induction structure.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-03: 1 request, 9,484 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.81 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.90 · compares with prior results 0.79 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Zero correlations offer a way to test the cuspidal structure of an automorphic L-function, but their leading self statistics retain only coarse degree and multiplicity data. We therefore adjoin a specified cuspidal L-function as a probe and study the resulting raw mixed correlations. Assuming a factorization into cuspidal L-functions over Q, we show that a mixed pair statistic detects the probe's occurrence and recovers its multiplicity. A suitable Gevrey-order-two test eliminates the smooth-density terms and gives a subexponential error. For arbitrary smooth plateau tests, the prime number theorem of Harcos and Thorner gives an error smaller than every fixed inverse power of log T, without a self-contragredience assumption. The stronger prime number theorem of Liu and Ye gives a subexponential remainder under its stated structural assumptions, with the possible norm-twist terms made explicit. To obtain arithmetic information beyond this overlap, we compute the correlation expansion at every fixed level and inverse-logarithmic order within the original strict small-support range. We treat the pair and higher correlations separately, using Jiang's rational-field fixed-data estimates to justify the required prime-power convergence. The pair calculation identifies the mixed arithmetic finite parts and gives detailed error estimates, including the contribution of norm twists. The higher-level calculation retains connected prime-power invariants, whose first cubic contribution appears at order (log T)^{-3}. Explicit third- and fourth-level formulas and examples explain the arithmetic content of these coefficients and the limitations of self-correlations. Varying powers of the probe separates their mixed components. These formulas, together with the distinct-index sieve, provide a collection of arithmetic probe responses toward the further problem of recovering automorphic induction structure.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Zero correlations offer a way to test the cuspidal structure of an automorphic L-function, but their leading self statistics retain only coarse degree and multiplicity data.	background 0.88	0.10	0.05	asserted 0.94
s1	We therefore adjoin a specified cuspidal L-function as a probe and study the resulting raw mixed correlations.	method 0.50	0.88	0.05	asserted 0.88
s2	Assuming a factorization into cuspidal L-functions over Q, we show that a mixed pair statistic detects the probe's occurrence and recovers its multiplicity.	result 0.97	0.92	0.48	asserted 1.00
s3	A suitable Gevrey-order-two test eliminates the smooth-density terms and gives a subexponential error.	method 0.87	0.72	0.07	asserted 1.00
s4	For arbitrary smooth plateau tests, the prime number theorem of Harcos and Thorner gives an error smaller than every fixed inverse power of log T, without a self-contragredience assumption.	result 0.73	0.15	0.05	asserted 1.00
s5	The stronger prime number theorem of Liu and Ye gives a subexponential remainder under its stated structural assumptions, with the possible norm-twist terms made explicit.	background 0.54	0.16	0.06	asserted 0.95
s6	To obtain arithmetic information beyond this overlap, we compute the correlation expansion at every fixed level and inverse-logarithmic order within the original strict small-support range.	method 0.58	0.89	0.08	asserted 0.98
s7	We treat the pair and higher correlations separately, using Jiang's rational-field fixed-data estimates to justify the required prime-power convergence.	method 0.96	0.68	0.04	asserted 1.00
s8	The pair calculation identifies the mixed arithmetic finite parts and gives detailed error estimates, including the contribution of norm twists.	result 0.90	0.91	0.12	asserted 1.00
s9	The higher-level calculation retains connected prime-power invariants, whose first cubic contribution appears at order (log T)^{-3}.	result 0.93	0.88	0.08	asserted 1.00
s10	Explicit third- and fourth-level formulas and examples explain the arithmetic content of these coefficients and the limitations of self-correlations.	contribution 0.73	0.88	0.11	asserted 1.00
s11	Varying powers of the probe separates their mixed components.	method 0.94	0.81	0.08	asserted 0.99
s12	These formulas, together with the distinct-index sieve, provide a collection of arithmetic probe responses toward the further problem of recovering automorphic induction structure.	future work 0.50	0.86	0.35	asserted 0.90

What kind of paper the classifier reports this is: theoretical result 0.86 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (27)

[18]

J. Liu and Y. Ye, Perron’s formula and the prime number theorem for automorphicL-functions, Pure Appl. Math. Q.3(2007), no. 2, 481–497. doi:10.4310/PAMQ.2007.v3.n2.a4

View on Pith doi:10.4310/pamq.2007.v3.n2.a4 2007
[26]

Y. Ye, Corrigendum to “Perron’s formula and the prime number theorem for automorphic L-functions,” submitted toPure Appl. Math. Q., 2026

View on Pith 2026
[1]

J. Arthur and L. Clozel,Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Mathematics Studies120, Princeton University Press, Princeton, NJ, 1989. doi:10.1515/9781400882403

View on Pith doi:10.1515/9781400882403 1989
[2]

E. Bogomolny, O. Bohigas, P. Leboeuf, and A. G. Monastra, On the spacing distribution of the Riemann zeros: corrections to the asymptotic result,J. Phys. A39(2006), 10743–10754. arXiv:math/0602270

View on Pith Pith review arXiv 2006
[3]

E. B. Bogomolny and J. P. Keating, Random matrix theory and the Riemann zeros I: three- and four-point correlations,Nonlinearity8(1995), no. 6, 1115–1131. doi:10.1088/0951-7715/8/6/013

View on Pith doi:10.1088/0951-7715/8/6/013 1995
[4]

E. B. Bogomolny and J. P. Keating, Random matrix theory and the Riemann zeros II:n-point correlations,Nonlinearity9(1996), no. 4, 911–935. doi:10.1088/0951-7715/9/4/006

View on Pith doi:10.1088/0951-7715/9/4/006 1996
[5]

F. Brumley, J. Thorner, and A. Zaman, with an appendix by C. J. Bushnell and G. Henniart, Zeros of Rankin–SelbergL-functions at the edge of the critical strip,J. Eur. Math. Soc.24 (2022), 1471–1541. doi:10.4171/JEMS/1134. 37 Yangbo Ye Zero correlations and automorphic induction

View on Pith doi:10.4171/jems/1134 2022
[6]

J. B. Conrey and N. C. Snaith, Applications of theL-functions ratios conjectures,Proc. London Math. Soc.(3)94(2007), 594–646. doi:10.1112/plms/pdl021

View on Pith doi:10.1112/plms/pdl021 2007
[7]

J. B. Conrey and N. C. Snaith, Triple correlation of the Riemann zeros,J. Théor. Nombres Bordeaux20(2008), no. 1, 61–106. doi:10.5802/jtnb.616; arXiv:math/0610495

View on Pith Pith review arXiv 2008
[8]

J.B.ConreyandN.C.Snaith, CorrelationsofeigenvaluesandRiemannzeros,Commun. Number Theory Phys.2(2008), no. 3, 477–536. doi:10.4310/CNTP.2008.v2.n3.a1; arXiv:0803.2795

View on Pith Pith review arXiv 2008
[9]

K. Ford and A. Zaharescu, Unnormalized differences between zeros ofL-functions,Compos. Math.151(2015), no. 2, 230–252. doi:10.1112/S0010437X14007659; arXiv:1305.2520

View on Pith Pith review arXiv 2015
[10]

G. Harcos and J. Thorner, A new zero-free region for Rankin–SelbergL-functions,J. Reine Angew. Math.822(2025), 179–201. doi:10.1515/crelle-2025-0009

View on Pith doi:10.1515/crelle-2025-0009 2025
[11]

D. A. Hejhal, On the triple correlation of zeros of the zeta function,Int. Math. Res. Not.1994 (1994), no. 7, 293–302. doi:10.1155/S1073792894000334

View on Pith doi:10.1155/s1073792894000334 1994
[12]

P. Humphries, with an appendix by F. Brumley, Standard zero-free regions for Rankin–Selberg L-functions via sieve theory,Math. Z.292(2019), 1105–1122. doi:10.1007/s00209-018-2136-8

View on Pith doi:10.1007/s00209-018-2136-8 2019
[13]

H. Iwaniec, W. Luo and P. Sarnak, Low lying zeros of families ofL-functions,Publ. Math. Inst. Hautes Études Sci.91(2000), 55–131. doi:10.1007/BF02698741; arXiv:math/9901141

View on Pith Pith review arXiv 2000
[14]

Y. Jiang, On Hypothesis H of Rudnick and Sarnak, preprint, arXiv:2507.20653v1. We use the rational-field fixed-data estimates in Theorem 1.3 and Section 5.1, as specified in Section 6

View on Pith Pith review arXiv
[15]

N. M. Katz and P. Sarnak,Random Matrices, Frobenius Eigenvalues, and Monodromy, Amer. Math. Soc. Colloq. Publ.45, American Mathematical Society, Providence, RI, 1999. doi:10.1090/coll/045

View on Pith doi:10.1090/coll/045 1999
[16]

N. M. Katz and P. Sarnak, Zeroes of zeta functions and symmetry,Bull. Amer. Math. Soc. (N.S.)36(1999), no. 1, 1–26. doi:10.1090/S0273-0979-99-00766-1

View on Pith doi:10.1090/s0273-0979-99-00766-1 1999
[17]

J. Liu and Y. Ye, Superposition of zeros of distinctL-functions,Forum Math.14(2002), 419–455

View on Pith 2002
[19]

J. Liu and Y. Ye, Correlation of zeros of automorphicL-functions,Sci. China Ser. A51(2008), 1147–1166

View on Pith 2008
[20]

J. Liu and Y. Ye, Functoriality of automorphicL-functions through their zeros,Sci. China Ser. A52(2009), 1–16. doi:10.1007/s11425-008-0088-x

View on Pith doi:10.1007/s11425-008-0088-x 2009
[21]

H. L. Montgomery, The pair correlation of zeros of the zeta function, inAnalytic Number Theory(St. Louis, MO, 1972), Proc. Sympos. Pure Math.24, American Mathematical Society, Providence, RI, 1973, pp. 181–193. doi:10.1090/pspum/024/9944

View on Pith doi:10.1090/pspum/024/9944 1972
[22]

A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function,Math. Comp. 48(1987), no. 177, 273–308. doi:10.1090/S0025-5718-1987-0866115-0. 38 Yangbo Ye Zero correlations and automorphic induction

View on Pith doi:10.1090/s0025-5718-1987-0866115-0 1987
[23]

B. Rodgers, Macroscopic pair correlation of the Riemann zeroes for smooth test functions,Q. J. Math.64(2013), no. 4, 1197–1219. doi:10.1093/qmath/has024; arXiv:1203.3275

View on Pith Pith review arXiv 2013
[24]

Z. Rudnick and P. Sarnak, Zeros of principalL-functions and random matrix theory,Duke Math. J.81(1996), 269–322. doi:10.1215/S0012-7094-96-08115-6

View on Pith doi:10.1215/s0012-7094-96-08115-6 1996
[25]

K. H. Wang and Y. Ye, From raw to distinct-index zero correlations for automorphicL-functions, working paper, 2026

View on Pith 2026
[27]

Y. Ye, Corrigendum to “Superposition of zeros of distinctL-functions,” submitted toForum Math., 2026. 39

View on Pith 2026

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [18]

J. Liu and Y. Ye, Perron’s formula and the prime number theorem for automorphicL-functions, Pure Appl. Math. Q.3(2007), no. 2, 481–497. doi:10.4310/PAMQ.2007.v3.n2.a4

View on Pith doi:10.4310/pamq.2007.v3.n2.a4 2007

[2] [26]

Y. Ye, Corrigendum to “Perron’s formula and the prime number theorem for automorphic L-functions,” submitted toPure Appl. Math. Q., 2026

View on Pith 2026

[3] [1]

J. Arthur and L. Clozel,Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Mathematics Studies120, Princeton University Press, Princeton, NJ, 1989. doi:10.1515/9781400882403

View on Pith doi:10.1515/9781400882403 1989

[4] [2]

E. Bogomolny, O. Bohigas, P. Leboeuf, and A. G. Monastra, On the spacing distribution of the Riemann zeros: corrections to the asymptotic result,J. Phys. A39(2006), 10743–10754. arXiv:math/0602270

View on Pith Pith review arXiv 2006

[5] [3]

E. B. Bogomolny and J. P. Keating, Random matrix theory and the Riemann zeros I: three- and four-point correlations,Nonlinearity8(1995), no. 6, 1115–1131. doi:10.1088/0951-7715/8/6/013

View on Pith doi:10.1088/0951-7715/8/6/013 1995

[6] [4]

E. B. Bogomolny and J. P. Keating, Random matrix theory and the Riemann zeros II:n-point correlations,Nonlinearity9(1996), no. 4, 911–935. doi:10.1088/0951-7715/9/4/006

View on Pith doi:10.1088/0951-7715/9/4/006 1996

[7] [5]

F. Brumley, J. Thorner, and A. Zaman, with an appendix by C. J. Bushnell and G. Henniart, Zeros of Rankin–SelbergL-functions at the edge of the critical strip,J. Eur. Math. Soc.24 (2022), 1471–1541. doi:10.4171/JEMS/1134. 37 Yangbo Ye Zero correlations and automorphic induction

View on Pith doi:10.4171/jems/1134 2022

[8] [6]

J. B. Conrey and N. C. Snaith, Applications of theL-functions ratios conjectures,Proc. London Math. Soc.(3)94(2007), 594–646. doi:10.1112/plms/pdl021

View on Pith doi:10.1112/plms/pdl021 2007

[9] [7]

J. B. Conrey and N. C. Snaith, Triple correlation of the Riemann zeros,J. Théor. Nombres Bordeaux20(2008), no. 1, 61–106. doi:10.5802/jtnb.616; arXiv:math/0610495

View on Pith Pith review arXiv 2008

[10] [8]

J.B.ConreyandN.C.Snaith, CorrelationsofeigenvaluesandRiemannzeros,Commun. Number Theory Phys.2(2008), no. 3, 477–536. doi:10.4310/CNTP.2008.v2.n3.a1; arXiv:0803.2795

View on Pith Pith review arXiv 2008

[11] [9]

K. Ford and A. Zaharescu, Unnormalized differences between zeros ofL-functions,Compos. Math.151(2015), no. 2, 230–252. doi:10.1112/S0010437X14007659; arXiv:1305.2520

View on Pith Pith review arXiv 2015

[12] [10]

G. Harcos and J. Thorner, A new zero-free region for Rankin–SelbergL-functions,J. Reine Angew. Math.822(2025), 179–201. doi:10.1515/crelle-2025-0009

View on Pith doi:10.1515/crelle-2025-0009 2025

[13] [11]

D. A. Hejhal, On the triple correlation of zeros of the zeta function,Int. Math. Res. Not.1994 (1994), no. 7, 293–302. doi:10.1155/S1073792894000334

View on Pith doi:10.1155/s1073792894000334 1994

[14] [12]

P. Humphries, with an appendix by F. Brumley, Standard zero-free regions for Rankin–Selberg L-functions via sieve theory,Math. Z.292(2019), 1105–1122. doi:10.1007/s00209-018-2136-8

View on Pith doi:10.1007/s00209-018-2136-8 2019

[15] [13]

H. Iwaniec, W. Luo and P. Sarnak, Low lying zeros of families ofL-functions,Publ. Math. Inst. Hautes Études Sci.91(2000), 55–131. doi:10.1007/BF02698741; arXiv:math/9901141

View on Pith Pith review arXiv 2000

[16] [14]

Y. Jiang, On Hypothesis H of Rudnick and Sarnak, preprint, arXiv:2507.20653v1. We use the rational-field fixed-data estimates in Theorem 1.3 and Section 5.1, as specified in Section 6

View on Pith Pith review arXiv

[17] [15]

N. M. Katz and P. Sarnak,Random Matrices, Frobenius Eigenvalues, and Monodromy, Amer. Math. Soc. Colloq. Publ.45, American Mathematical Society, Providence, RI, 1999. doi:10.1090/coll/045

View on Pith doi:10.1090/coll/045 1999

[18] [16]

N. M. Katz and P. Sarnak, Zeroes of zeta functions and symmetry,Bull. Amer. Math. Soc. (N.S.)36(1999), no. 1, 1–26. doi:10.1090/S0273-0979-99-00766-1

View on Pith doi:10.1090/s0273-0979-99-00766-1 1999

[19] [17]

J. Liu and Y. Ye, Superposition of zeros of distinctL-functions,Forum Math.14(2002), 419–455

View on Pith 2002

[20] [19]

J. Liu and Y. Ye, Correlation of zeros of automorphicL-functions,Sci. China Ser. A51(2008), 1147–1166

View on Pith 2008

[21] [20]

J. Liu and Y. Ye, Functoriality of automorphicL-functions through their zeros,Sci. China Ser. A52(2009), 1–16. doi:10.1007/s11425-008-0088-x

View on Pith doi:10.1007/s11425-008-0088-x 2009

[22] [21]

H. L. Montgomery, The pair correlation of zeros of the zeta function, inAnalytic Number Theory(St. Louis, MO, 1972), Proc. Sympos. Pure Math.24, American Mathematical Society, Providence, RI, 1973, pp. 181–193. doi:10.1090/pspum/024/9944

View on Pith doi:10.1090/pspum/024/9944 1972

[23] [22]

A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function,Math. Comp. 48(1987), no. 177, 273–308. doi:10.1090/S0025-5718-1987-0866115-0. 38 Yangbo Ye Zero correlations and automorphic induction

View on Pith doi:10.1090/s0025-5718-1987-0866115-0 1987

[24] [23]

B. Rodgers, Macroscopic pair correlation of the Riemann zeroes for smooth test functions,Q. J. Math.64(2013), no. 4, 1197–1219. doi:10.1093/qmath/has024; arXiv:1203.3275

View on Pith Pith review arXiv 2013

[25] [24]

Z. Rudnick and P. Sarnak, Zeros of principalL-functions and random matrix theory,Duke Math. J.81(1996), 269–322. doi:10.1215/S0012-7094-96-08115-6

View on Pith doi:10.1215/s0012-7094-96-08115-6 1996

[26] [25]

K. H. Wang and Y. Ye, From raw to distinct-index zero correlations for automorphicL-functions, working paper, 2026

View on Pith 2026

[27] [27]

Y. Ye, Corrigendum to “Superposition of zeros of distinctL-functions,” submitted toForum Math., 2026. 39

View on Pith 2026