REVIEW
Large values of Hardy's Function from the Selberg class
Number Theory · Submitted 27 Oct 2026 · 37 references
Pith's verdict: Solid, competent extension of the resonance method for Hardy's Z to degree-1 and degree-2 Selberg-class L-functions; the degree-2 unconditional results for cusp forms, Dedekind zeta, and products are new and correctly argued.
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Referee report
The paper
The paper claims
Hardy's function for degree-1 and many degree-2 L-functions attains values exp(C sqrt(log T / log log T)).
The reading
What carries the argument
The resonance method with a Dirichlet polynomial resonator R(t)=∑_{m≤N} r(m) $m^{{-it}}$ where r(n)=a_F(n) f(n) and f is a multiplicative function supported on squarefree integers in a carefully chosen range. The lower bound for the integral of F(1/2+it)|R(t)|^2 comes from Euler product calculations and the Selberg normality conjecture (or its verified versions), while the upper bound for the integral of Z_F(t)|R(t)|^2 is obtained by shifting the contour and using exponential sum estimates (Hypothesis 1 for degree 2, or faster derivative estimates for degree 1, or special handling for Dedekind zeta and products). The balance of these two integrals forces large values of Z_F(t).
Extended reading notes
Core claim
The paper establishes that for any L-function F of degree 1 in the Selberg class (including shifts of Dirichlet L-functions) and for several degree 2 functions (Hecke cusp forms, Maass cusp forms, Dedekind zeta of quadratic fields, products L(s,χ1)L(s,χ2) with q1q2 not a square), we have max_{T≤t≤2T} ± Z_F(t) ≫ exp(C_F √(log T / log log T)) where C_F is explicit: 1/2 for degree 1, 1/√2 for cusp forms and the listed special degree-2 cases, and √(κ(1-γ)) for general degree 2 under an exponential sum hypothesis and the Selberg normality conjecture. The proof is based on Soundararajan's resonance method with a Dirichlet polynomial resonator, and requires exponential sum estimates for the Dirichlet coefficients. The paper also obtains an unconditional result for Maass cusp forms with constant √(25/71) using the Kim-Sarnak bound.
Load-bearing premise
For the general degree-2 result (Theorem 1.3), the load-bearing premise is Hypothesis 1: there exists γ<1 such that the exponential sum ∑_{n≤x} a_F(n) $e^{{2π i n x}}$ is O(x^γ (log x)^{γ1}) uniformly. This hypothesis is not proved for all Selberg class degree-2 functions and is only verified for specific families (cusp forms, etc.). If γ were 1, the condition α<1-γ would require α<0, making the resonator useless.
Editorial extensions
If this is right
- For all degree-1 Selberg class L-functions, Hardy's function has maximal values of order exp((1/2-ϵ)√(log T / log log T)), confirming the resonance method works universally for this degree.
- For holomorphic and Maass cusp forms of arbitrary level and nebentypus, the same lower bound with constant 1/√2 is obtained, providing the first such result for these L-functions.
- For Dedekind zeta of quadratic fields and for products of distinct primitive Dirichlet L-functions with non-square conductor product, the constant is also 1/√2, showing that even when a uniform exponential sum bound is not available, the method can still succeed via alternative estimates.
- The method fails for degree ≥3, so a new idea is needed to extend such results to higher degree.
- The results contribute to the understanding of the distribution of values of L-functions on the critical line and support the consistency of the Selberg class framework.
Reading between the lines
- The explicit constants C_F appear to be maximal achievable under current techniques: for degree 1 the constant 1/2 matches the exponent from the Riemann zeta case, suggesting an optimal bound for this method; for degree 2 the constant 1/√2 is the same across several families, possibly indicating a universal bound for degree 2 L-functions under suitable hypotheses.
- The reliance on the unproven Hypothesis 1 for general degree-2 L-functions highlights a gap: the exponential sum bound may be a barrier to proving such large values for all degree-2 Selberg class functions.
- The treatment of Dedekind zeta and products via Lemmas 5.2 and 5.3 provides a template for handling L-functions that do not admit a uniform exponential sum bound but still have enough cancellation from Diophantine approximation.
- The results suggest a connection between the constant C_F and the rank (degree) of the L-function: the method naturally gives a constant that decreases as the degree increases, consistently with the difficulty of controlling the integral of Z_F(t)R(t)^2 for higher degree.
Original abstract
Let $Z_F(t):=F(1/2+it) H(1/2+it)^{-1/2}$ denote Hardy's function, where the function $F(s)$ belongs to the Selberg class with polynomial Euler product and satisfies the functional equation $F(s)=H(s)\overline{F}(1-\bar{s})$. We prove that for every function of degree $1$ and for certain functions of degree $2$ belonging to the Selberg class, one has \begin{align*} \max_{T\leq t\leq 2T} \pm Z_F(t) &\gg \exp\left(C_F \sqrt{ \frac{{\log T}}{{\log \log T}} } \right), \end{align*} where the constant $C_F$ depends on $F(s)$ and are explicitly determined.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Let $Z_F(t):=F(1/2+it) H(1/2+it)^{-1/2}$ denote Hardy's function, where the function $F(s)$ belongs to the Selberg class with polynomial Euler product and satisfies the functional equation $F(s)=H(s)\overline{F}(1-\bar{s})$. We prove that for every function of degree $1$ and for certain functions of degree $2$ belonging to the Selberg class, one has \begin{align*} \max_{T\leq t\leq 2T} \pm Z_F(t) &\gg \exp\left(C_F \sqrt{ \frac{{\log T}}{{\log \log T}} } \right), \end{align*} where the constant $C_F$ depends on $F(s)$ and are explicitly determined.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $Z_F(t):=F(1/2+it) H(1/2+it)^{-1/2}$ denote Hardy's function, where the function $F(s)$ belongs to the Selberg class with polynomial Euler product and satisfies the functional equation $F(s)=H(s)\overline{F}(1-\bar{s})$. | background 1.00 | 0.12 | 0.02 | asserted 0.80 |
| s1 | We prove that for every function of degree $1$ and for certain functions of degree $2$ belonging to the Selberg class, one has \begin{align*} \max_{T\leq t\leq 2T} \pm Z_F(t) &\gg \exp\left(C_F \sqrt{ \frac{{\log T}}{{\log \log T}} } \right), \end{align*} where the constant $C_F$ depends on $F(s)$ and are explicitly determined. | result 0.99 | 0.95 | 0.95 | asserted 1.00 |
What was examined, and what was not
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References
Discussion
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