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arxiv: 2604.07474 · v1 · submitted 2026-04-08 · 🧮 math.NT

On Lower Bounds for sums of Fourier Coefficients of Twist-Inequivalent Newforms

Moni Kumari , Prabhat Kumar Mishra , Jyotirmoy Sengupta This is my paper

Pith reviewed 2026-05-10 17:06 UTC · model grok-4.3

classification 🧮 math.NT
keywords newformsFourier coefficientslargest prime factorBrun's sieveHecke eigenvaluestwist-inequivalentGeneralized Riemann Hypothesismultiplicity one
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The pith

For twist-inequivalent non-CM newforms with integer Fourier coefficients, the sum a_f(p) + a_g(p) has largest prime factor larger than (log p)^{1/14} (log log p)^{3/7-ε} for almost all primes p.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes lower bounds on the largest prime factor of sums of p-th Fourier coefficients from two twist-inequivalent non-CM normalized newforms that have integer coefficients. The bound exceeds (log p) to the 1/14 power times (log log p) to the 3/7 minus epsilon power for almost all primes p. The authors extend the result via Brun's sieve to a positive-density set of all positive integers. Under the generalized Riemann hypothesis the absolute value of the sum grows exponentially with p. They also show that if the sum stays small on a positive-density set of primes then the two forms must be twist-equivalent.

Core claim

The central claim is that for twist-inequivalent non-CM normalized newforms f and g with integer Fourier coefficients, the largest prime factor P of a_f(p) + a_g(p) satisfies P(a_f(p) + a_g(p)) > (log p)^{1/14} (log log p)^{3/7-ε} for almost all primes p and any ε > 0. Beyond primes, Brun's sieve yields the same phenomenon for a set of positive integers with natural density one. Under the generalized Riemann hypothesis the absolute value |a_f(p) + a_g(p)| grows exponentially with p. If a_f(p) + a_g(p) remains small on a positive-density subset of primes then f and g are twist-equivalent by a quadratic character.

What carries the argument

The sums a_f(p) + a_g(p) of Hecke eigenvalues at primes p for twist-inequivalent non-CM newforms, controlled through analytic properties of the associated L-functions and sieved for large prime factors.

Load-bearing premise

The newforms are twist-inequivalent, non-CM, normalized, and possess integer Fourier coefficients.

What would settle it

Finding two twist-inequivalent non-CM newforms with integer coefficients such that a_f(p) + a_g(p) factors completely into primes smaller than (log p)^{1/14} (log log p)^{3/7} for a positive proportion of primes p would disprove the main lower bound.

read the original abstract

In this article, we address the lower bounds for the sums $a_f(p)+a_g(p)$ of the $p$-th Fourier coefficients of two twist-inequivalent, non-CM normalized newforms $f$ and $g$. Our main result shows that for such forms with integer Fourier coefficients, the largest prime factor of $a_f(p)+a_g(p)$ satisfies $P(a_f(p)+a_g(p)) > (\log p)^{1/14} (\log \log p)^{3/7-\epsilon}$ for almost all primes $p$ and for any $\epsilon > 0$. Beyond primes, we apply Brun's sieve to show that a similar phenomenon holds for a set of positive integers with natural density one. The main result is further strengthened under the Generalized Riemann Hypothesis, where we establish exponential growth for the absolute value of $a_f(p)+a_g(p)$ in terms of $p$.Additionally, we derive an interesting result related to the multiplicity one theorem, demonstrating that if the sum $a_f(p)+a_g(p)$ is small for a positive-density subset of primes, then $f$ and $g$ must be twist-equivalent by a quadratic character.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper proves lower bounds on the largest prime factor P(a_f(p) + a_g(p)) for the sum of p-th Fourier coefficients of two twist-inequivalent non-CM normalized newforms f and g with integer coefficients. The main theorem asserts P(a_f(p) + a_g(p)) > (log p)^{1/14} (log log p)^{3/7-ε} for almost all primes p and any ε > 0. It extends the result to a positive-density set of integers via Brun's sieve, obtains exponential growth of |a_f(p) + a_g(p)| under GRH, and shows that if a_f(p) + a_g(p) is small on a positive-density set of primes then f and g must be twist-equivalent.

Significance. If the error-term tracking and exceptional-set handling hold, the result supplies an explicit, unconditional lower bound on the prime factors of sums of Hecke eigenvalues, obtained from equidistribution, zero-density estimates, and Brun's sieve. The GRH strengthening and the multiplicity-one corollary are natural consequences of the same machinery and add value. The work is grounded in standard analytic properties of L-functions attached to newforms and does not introduce new conjectures.

major comments (2)
  1. [§3 (Brun sieve application)] The abstract and introduction claim the exponent 1/14 arises from Brun's sieve applied to the level of distribution coming from large-sieve inequalities for the Rankin-Selberg L-functions L(f ⊗ g, s) and the individual L(f, s), L(g, s). The precise optimization (including the dependence on the conductor and the ε-loss) must be written out explicitly, as the current sketch leaves open whether the 3/7-ε in the log-log term survives after removing the exceptional set of density o(1).
  2. [§4 (GRH strengthening)] Under GRH the paper asserts exponential growth |a_f(p) + a_g(p)| ≫ p^δ for some δ > 0 on a positive-density set of p. The derivation should record the explicit δ obtained from the GRH-implied zero-free region and the resulting lower bound on the sum; without this the claim remains qualitative.
minor comments (3)
  1. [Theorem 1.1] The statement 'for almost all primes p' should be quantified with an explicit density (e.g., all but O(x / log^A x) primes up to x) so that the exceptional set is visible for applications.
  2. [Introduction] Notation for the largest prime factor P(n) and the sum S(p) = a_f(p) + a_g(p) should be fixed once at the beginning and used consistently; occasional switches between a_f(p) + a_g(p) and the symbol S(p) slow reading.
  3. [§5] The multiplicity-one corollary (if S(p) is small on a positive-density set then f and g are twist-equivalent) is interesting but its proof sketch relies on the same sieve; a short self-contained argument or reference to the precise density estimate used would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. We address the two major comments below and will revise the manuscript accordingly to incorporate the requested explicit details.

read point-by-point responses

  1. Referee: [§3 (Brun sieve application)] The abstract and introduction claim the exponent 1/14 arises from Brun's sieve applied to the level of distribution coming from large-sieve inequalities for the Rankin-Selberg L-functions L(f ⊗ g, s) and the individual L(f, s), L(g, s). The precise optimization (including the dependence on the conductor and the ε-loss) must be written out explicitly, as the current sketch leaves open whether the 3/7-ε in the log-log term survives after removing the exceptional set of density o(1).

    Authors: We agree that the optimization must be presented in full detail. In the revised manuscript we will add an explicit computation of the level of distribution obtained from the large-sieve inequalities for L(f⊗g,s), L(f,s) and L(g,s), including the dependence on the conductors of f and g and the precise ε-losses incurred. We will then verify that, after excising the exceptional set of density o(1), the lower bound P(a_f(p)+a_g(p)) ≫ (log p)^{1/14} (log log p)^{3/7-ε} continues to hold for every ε>0. revision: yes

  2. Referee: [§4 (GRH strengthening)] Under GRH the paper asserts exponential growth |a_f(p) + a_g(p)| ≫ p^δ for some δ > 0 on a positive-density set of p. The derivation should record the explicit δ obtained from the GRH-implied zero-free region and the resulting lower bound on the sum; without this the claim remains qualitative.

    Authors: We accept the suggestion. In the revised version we will derive and state the explicit positive constant δ that follows from the GRH zero-free region for the Rankin-Selberg L-function L(f⊗g,s) (and the individual L-functions), together with the resulting lower bound on |a_f(p)+a_g(p)| that holds on a positive-density set of primes p. This will render the exponential growth quantitative. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives its lower bound on the largest prime factor of a_f(p) + a_g(p) by applying Brun's sieve to the distribution of Hecke eigenvalues for twist-inequivalent non-CM newforms, relying on standard zero-density estimates and analytic properties of the associated L-functions. These are external, established tools independent of the target result. The multiplicity-one implication is presented as a derived consequence rather than an input assumption. No step reduces the claimed bound to a fitted parameter, self-definition, or load-bearing self-citation chain. The argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the complete list of background results invoked in the proofs cannot be audited. The visible assumptions are standard in the field.

axioms (2)
  • domain assumption Generalized Riemann Hypothesis for the L-functions attached to the newforms
    Invoked to obtain the exponential lower bound on |a_f(p) + a_g(p)|.
  • standard math Standard analytic continuation, functional equation, and Euler-product properties of newform L-functions
    Required for the application of sieve methods to the coefficient sums.

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

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    Kumar and M

    MR424730 ↑12 [KK23] A. Kumar and M. Kumari,Divisors of Fourier coefficients of two newforms, Pacific J. Math.326 (2023), no. 1, 85–107. MR4678150↑6, 7, 8 [LMO79] J. C. Lagarias, H. L. Montgomery, and A. M. Odlyzko,A bound for the least prime ideal in the Chebotarev density theorem, Invent. Math.54(1979), no. 3, 271–296. MR553223↑4 [LO77] J. C. Lagarias an...

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