A zero-density estimate for the symmetric square $L$-functions in the weight aspect

Songchen Yin

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A zero-density estimate for the symmetric square $L$-functions in the weight aspect

Songchen Yin

Number Theory · Submitted 27 Oct 2026

Pith's verdict: Plausible first weight-aspect zero-density estimate for the symmetric square, built on Khan-Hough machinery; the §3.4 cancellation is the place to referee, and the stated t-range needs a one-sentence repair.

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arXiv ID2610.37371 (version 2)

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CitationSongchen Yin. "A zero-density estimate for the symmetric square $L$-functions in the weight aspect." arXiv:2610.37371 (2026). https://pith.science/paper/PALSED3T

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Editor's note

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Two things to know. The result is genuinely new: no one has a weight-aspect zero-density estimate for sym^2, and the adaptation from Hough's L(s,f) program is substantial, not a routine rerun. The Hecke relation with λ_f(r^2), the shifted Khan second moment, the polynomial mollifier, and the Selberg zero-counting step are all coherent, and the numerical bookkeeping is careful enough that I did not find an obvious exponent failure. If the proof is right, this is a useful tool, though it stays inside analytic number theory.

The soft spot is exactly where the reader and the stress-test note point: §3.4. The comparison of the two evaluations of the twisted second moment is the load-bearing step, and it is compressed into one paragraph. What needs to be displayed is the second evaluation itself: that pairing λ_f(m^2) with λ_f(r^2) really only flips the outer (d1d2)^{it}, with no extra residue and no change of K^{2it} to K^{-2it}. I could not certify that from the text alone, and the referee should ask for it in full.

I disagree with one part of the stress-test note: the Lemma 7 mismatch is not a real hole. The comparison controls Re(K^{2-2σ+2it}ΣJ sin); multiply by K^{2σ-2}, write h = 2iΣJ sin, and the bound becomes Im(K^{2it}h) = O(f). Lemma 7 then applies with K replaced by K^2. That is a rewrite, not a missing argument. The unexpanded second evaluation is the part that matters.

There is also a smaller issue: Theorem 1 allows t2 < K^{1/5}, while Proposition 2 only covers |t| ≤ K^δ with δ < 1/5. If t1,t2 are fixed, that is harmless for large K; if they are meant to grow with K, the stated range is not covered. The fix is either to prove Proposition 2 at δ = 1/5 or to state the theorem with t2 < K^δ. Someone should also note that Lemma numbering jumps to 11, and the “By Python” Kummer identity is explicit but would be better with a short proof or supplied verification.

Bottom line: send it to a serious referee, with instructions to write out §3.4 and to fix the t-range. If the second evaluation checks out, this is a solid first result and worth citing. My own verdict is conditional, but refereeable now.

What would settle it

Take a concrete twist, say $r=2$ with $\sigma=3/4$ and $t=1$, and compute the two off-diagonal expressions $(OD_1)+(OD_2)$ and $(OD'_1)+(OD'_2)$ directly from the definitions of $I_\pm$ and $\tilde I_\pm$: the proof requires their difference to equal $2i\sum_{d\mid r^2}\Re(K^{2-2\sigma+2it}J(t,d_1,d_2)\sin(t\log d_1d_2))$, bounded by $R'=K^{-1+\epsilon}|t|^{1+\sigma}r+K^{1-2\sigma+\epsilon}|t|^{2\sigma}r^\sigma$. A direct symbolic or numerical check of whether $\sum_{d\mid r^2}J(t,d_1,d_2)(d_1d_2)^{it}$ really equals its image under $t\mapsto -t$ would settle the point: any discrepancy of size $K^{2\sigma-2}$ that does not vanish along the geometric sequence used in Lemma 7 breaks the symmetry, changes the surviving residue $S_{OD}$, and invalidates the rate $K^{1-(1/19)(\sigma-1/2)}$.

Referee report

Major objections (3)

[§3.4, comparison of the two evaluations] The conclusion that the difference of the two evaluations of the twisted second moment is O(R') is the hinge of the proof of Proposition 1, but the second evaluation is not actually carried out. The manuscript asserts that pairing lambda_f(m^2) with lambda_f(r^2) leaves the diagonal term unchanged, changes only the factor (d1 d2)^{it} in the off-diagonal term, and leaves the phase K^{2-2sigma+2it} and the zeta factors zeta(1+2x+4it) unchanged. None of the intervening residues and error terms of the second evaluation is displayed, and the sentence 'since we are calculating the same second moment' does not by itself rule out an extra residue or a term of size comparable to the S_{alpha,pm} contributions. Because S_OD in Proposition 1 is obtained from this comparison, an uncancelled term would change S_OD by a main term and the exponent 1/19 in Theorem 1 would not follow. The authors should display both evaluations in parallel and bound the difference explicitly, including the remainder terms from the approximate functional equation.
[§3.4, Lemma 7] The application of Lemma 7 is formally mismatched. Lemma 7 concludes g(t)=0 from Im(K^{it} g(t)) << f(K,t) with g independent of K. The displayed comparison gives Re(K^{2-2sigma+2it} sum_{d|r^2} J(t,d_1,d_2)((d_1d_2)^{it}-(d_1d_2)^{-it})) = O(R'), i.e., after division, a bound on Re(K^{2it} G(t)) with G(t)=sum_{d|r^2} J(t,d_1,d_2)((d_1d_2)^{it}-(d_1d_2)^{-it}), not on Im(K^{it}g(t)). A real-part variant with the K^{2it} phase can likely be proved by the same phase-sampling idea, but it must be stated, and one must verify that G(t) does not depend on K. Moreover, Lemma 7 is phrased for a fixed t as K tends to infinity; if t is allowed to depend on K through t <= K^delta, the proof of the lemma no longer applies directly. This is a load-bearing formal gap in the derivation of sum_d J(t,d_1,d_2)(d_1d_2)^{it} = sum_d J(t,d_1,d_2)(d_1d_2)^{-it}.
[§3.4, error terms of the comparison] The phrase 'the difference ... can be bounded by the error term arising from the off-diagonal term' leaves unquantified which of the error terms from the two evaluations are controlled. The remainders in Proposition 1 include K^epsilon r^{2sigma-1}, which for fixed r does not tend to zero as K tends to infinity. If such a term survives in the difference of the two evaluations rather than cancelling, then the subsequent choice f(K,t)=R'K^{2sigma-2} in Lemma 7 would not tend to zero, and the symmetry conclusion would fail. The comparison therefore needs an explicit accounting of every term of both evaluations, not only a statement that the two calculations agree because they compute the same second moment.

Minor objections (4)

[Global] The lemma numbering jumps from Lemma 8 to Lemma 11; the lemmas after Lemma 8 should be renumbered consecutively.
[Throughout] There are several typos and infelicities: 'satisfiy', 'trivival', 'Fsatiefies', 'seperates', and 'we follows Lemma 3.2'. These should be corrected in a revision.
[§4, definition of hat F(s)] The identity {}_1F_1(7;14;b)=... is introduced with 'By Python we have'. Since this identity is the basis for the Mellin-transform bounds on the mollifier, it should be stated as an exact finite computation with a brief explanation or reference, rather than attributed to a Python computation.
[Proposition 1, displayed S_OD] The displayed formula for S_OD has unbalanced parentheses, which makes the argument of the Mellin transform and the zeta factors hard to parse. Equation labels for the main displays in Sections 3 and 4 would substantially improve the readability.

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The paper

The paper claims

Symmetric-square L-functions of large weight obey a zero-density estimate in the weight aspect: the averaged count of zeros with real part above σ is O((t2−t1)K^{1−(σ−1/2)/19} log K).

The reading

This paper proves a zero-density estimate for the symmetric square $L$-functions attached to Hecke eigencuspforms of large weight: averaged over even weights $k$ near $K$, the number of zeros with real part above $\sigma$ and imaginary part between $t_1$ and $t_2$ is $O((t_2-t_1)K^{1-(1/19)(\sigma-1/2)}\log K)$, provided $t_1,t_2

What carries the argument

Four mechanisms carry the argument. (1) The symmetric square $L$-function $L(s,\operatorname{sym}^2 f)=\zeta(2s)\sum_{n\ge1}\lambda_f(n^2)n^{-s}$, entire with functional equation $s\mapsto 1-s$, whose coefficients obey the Hecke relation $\lambda_f(n^2)\lambda_f(r^2)=\sum_{d\mid(n^2,r^2)}\lambda_f(n^2r^2/d^2)$. (2) The twisted second moment (Proposition 1): the harmonic average over $f\in H_k$ of $|L(\sigma+it,\operatorname{sym}^2 f)|^2\lambda_f(r^2)$, expanded through the approximate functional equation and the Petersson trace formula into a diagonal term plus Kloosterman sums; an averaged $J$-Bessel lemma (Lemma 4) converts the off-diagonal piece into explicit oscillatory integrals labeled $I_\pm(r)$. (3) The cancellation step: the same second moment evaluated twice yields off-diagonal integrals whose difference is $2i\sum_{d\mid r^2}\Re(K^{2-2\sigma+2it}J(t,d_1,d_2)\sin(t\log d_1d_2))=O(R')$; Lemma 7, running $K$ through a geometric sequence that rotates the phase, forces the underlying quantity to vanish, giving the symmetry $\sum J(t,d_1,d_2)(d_1d_2)^{it}=\sum J(t,d_1,d_2)(d_1d_2)^{-it}$ and leaving only the residue $S_{OD}$ at $\sigma-1/2$. (4) The mollifier $M(s,\operatorname{sym}^2 f)=\sum_m b_f(m)F(s(m))m^{-s}$, where $s(m)$ is the squarefree kernel and $F$ is a smooth cutoff of length $M=K^\theta$; its moments are evaluated by rewriting coefficient sums as Euler products and bounding shifted zeta ratios (Lemma 8). Lemma 11, the classical zero-counting identity, finally converts the moment bounds into the zero count.

Extended reading notes

Core claim

The paper's central claim is Theorem 1: for $0<t_1<t_2<K^{1/5}$ with $t_2-t_1>1/\log K$ and $1/2+1/\log K<\sigma<1$, the $\psi$-weighted harmonic average over even weights $k$ of the zero count $N(\sigma,t_1,t_2,\operatorname{sym}^2 f)$ is $O((t_2-t_1)K^{1-(1/19)(\sigma-1/2)}\log K)$. The route is the classical mollified-moment scheme: control zeros by showing that $L(\sigma+it,\operatorname{sym}^2 f)$ times a short Dirichlet polynomial (the mollifier) is close to $1$ on average. The new input is a twisted second moment asymptotic (Proposition 1) evaluating the harmonic average of $|L(\sigma+it,\operatorname{sym}^2 f)|^2\lambda_f(r^2)$ as a sum of three explicit main terms $S_+,S_-,S_{OD}$ plus an error. The off-diagonal part is evaluated by computing the same second moment twice, once pairing $\lambda_f(m^2)$ with $\lambda_f(r^2)$ and once pairing $\lambda_f(n^2)$ with $\lambda_f(r^2)$; a limiting-sequence lemma (Lemma 7) forces the two off-diagonal integrals to differ only by a term that vanishes, which yields a symmetry under $t\mapsto -t$ that cancels the poles at $\pm it$ and leaves the single residue $S_{OD}$ as the effective main term. From this, Proposition 2 shows the mollified product is $1$ up to a small error and that its harmonically averaged square is the natural main term plus $O(K^{1-(1/9)(\sigma-1/2)})$, which a classical zero-counting lemma (Lemma 11) converts into the count of Theorem 1.

Load-bearing premise

The load-bearing premise is the assertion in §3.4 that the two evaluations of the same twisted second moment have identical diagonal terms and off-diagonal difference $O(R')$; if any term currently absorbed in that error is in fact as large as $K^{2\sigma-2}$, Lemma 7's conclusion fails, the residue $S_{OD}$ picks up an extra main term, and the exponent $1/19$ in Theorem 1 does not follow.

Editorial extensions

If this is right

On average over even weights $k$ near $K$, zeros of $L(s,\operatorname{sym}^2 f)$ with real part $\ge\sigma$ number at most $O((t_2-t_1)K^{1-(1/19)(\sigma-1/2)}\log K)$ in any window $t_1<\gamma<t_2$, so 'most' zeros lie within $O(1/\log K)$ of the critical line.
The estimate is uniform for $\sigma-1/2$ down to $1/\log K$ and for imaginary parts up to $K^{1/5}$; the paper frames such zero-density control as the standard way to avoid the Riemann Hypothesis in applications.
Proposition 1 stands alone as a reusable asymptotic for the twisted second moment $\sum^h |L(\sigma+it,\operatorname{sym}^2 f)|^2\lambda_f(r^2)$ with explicit main terms, valid for twists $r<K^{1-\delta-\epsilon}$.
Proposition 2 shows the mollifier performs as the scheme requires: the mollified product is $1$ up to a small error, and the averaged squared moment is the natural main term plus $O(K^{1-(1/9)(\sigma-1/2)})$, the exact shape that yields the zero-density theorem.

Reading between the lines

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

The exponent $1/19$ comes from error-term bookkeeping — the $O(R')$ comparison in §3.4 and the convexity bound controlling $S_-$ — rather than from an intrinsic obstruction; a sharper accounting of the two-evaluation difference, ideally an exact asymptotic instead of an order-of-magnitude bound, should improve the constant, and that is the natural next refinement.
Lemma 7 is a transferable gadget: whenever one average is evaluated two ways and the two off-diagonal phases differ by rotation in $t\log(d_1d_2)$, the same geometric-sequence argument forces an exact symmetry; families with structured off-diagonal terms, such as level-aspect or higher-rank spectral families, are natural places to try the trick.
A testable extension is to push the imaginary-part range beyond $|t|<K^{1/5}$; the zeta-ratio bounds in Lemma 8 and the treatment of $S_-$ are the visible bottlenecks, so a stronger estimate for products of shifted zeta functions would extend the theorem's range with the same structure.

Original abstract

In this paper, we prove a zero-density estimate for the symmetric square $L$-functions associated with Hecke eigencuspforms in the weight aspect. The proofs are based on the mollified second moment of these $L$-functions.

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

In this paper, we prove a zero-density estimate for the symmetric square $L$-functions associated with Hecke eigencuspforms in the weight aspect. The proofs are based on the mollified second moment of these $L$-functions.

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s0	In this paper, we prove a zero-density estimate for the symmetric square $L$-functions associated with Hecke eigencuspforms in the weight aspect.	contribution 0.59	0.97	0.96	asserted 1.00
s1	The proofs are based on the mollified second moment of these $L$-functions.	method 1.00	0.84	0.03	asserted 1.00

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