Riesz Sum Analysis for Cusp Forms

• Riesz sums are smoothed partial sums of Fourier coefficients from cusp forms that reveal fine cancellation phenomena and error term behavior.
• They utilize analytic techniques such as trace formulas, oscillatory integrals, and spectral expansions to derive precise mean-square and moment estimates.
• Their study supports applications ranging from subconvexity and shifted convolution problems to insights in quantum unique ergodicity and automorphic L-functions.

A Riesz sum attached to a cusp form is a smoothed or weighted partial sum of the Fourier coefficients of an automorphic cusp form, designed to probe fine cancellation phenomena, mean value properties, and the oscillatory structure of these coefficients beyond what is possible with coarse (sharp cutoff) sums. Riesz summation techniques, originally arising in classical analysis, are in the modern analytic theory of cusp forms deeply intertwined with trace formulas, spectral expansions, and the paper of automorphic $L$-functions. Their analysis reveals mean square and higher moment behavior, uncovers connections with harmonic analysis, and controls error terms in a variety of problems spanning short sums, quadratic progressions, and shifted convolution problems.

1. Definitions and Canonical Forms

Let $$f(z) = \sum_{n=1}^\infty a(n) n^{(\kappa-1)/2} e(nz)$$ be a holomorphic cuspidal Hecke eigenform of even integer weight $\kappa$ for the full modular group. The classical "long" exponential sum

$S(M; \alpha) = \sum_{n=1}^M a(n)e(n\alpha)$

has been extensively studied. Riesz means refine this via smoothing or by restricting to nonstandard summation sets (short intervals, polynomial progressions, arithmetic progressions, spectral intervals, etc.) and inserting smooth/analytic weights. Canonical examples include:

• Short Exponential Sums:

$$S(x) = \sum_{n \in [x, x+V(x)]} a(n) e\left(\frac{hn}{k}\right)$$

for $V(x) \ll x^{1/2}$ and small denominator $k$ (Ernvall-Hytönen, 2010).

• Quadratic Progressions (Non-split Sums):

$$S(d, Y) = \sum_{n} a(n^2+d) W\left(\frac{n^2+d}{Y}\right)$$

with $W$ a smooth cutoff (Templier et al., 2011).

• Weighted (Smoothed) Sums:

$$R_f(X) = \sum_{n} a(n) \left(1 - \frac{n}{X}\right)^\delta$$

for some $\delta > 0$ (Riesz mean kernel), smoothing the sharp cutoff.

• Resonance/Riesz Sums with Oscillatory Twist:

$$S_X(f,g;\alpha,\beta) = \sum_{X \leq n \leq 2X} \lambda_f(n) \lambda_g(n) e(\alpha n^\beta) \phi\left(\frac{n}{X}\right)$$

for normalized coefficients $\lambda_f(n)$, $\phi$ smooth (Gillespie et al., 2022).

In all cases, the sum can be interpreted as a Riesz mean if the sharp cutoff is replaced by a smooth, compactly supported weight or by analytic regularization (e.g., exponential damping $e^{-s\sqrt{n}}$ (Liu, 2023)).

2. Analytic Methods: Mean Squares and Oscillatory Integrals

For sharp understanding of cancellation and moments, the mean square or $p$-th moment (for $p > 2$) of Riesz sums is a central object: $$\int_{M}^{M+A} \left| S(x) \right|^2 x^{1/2} w(x) dx$$ where $w$ is a smooth weight and $S(x)$ is as above (Ernvall-Hytönen, 2010). The key steps involve:

• Approximate Formulas: Expansions for $a(n)$ involving oscillatory main terms (e.g., cosine terms capturing the oscillating nature of the coefficients).
• Diagonal and Off-Diagonal Decomposition: Write $|S(x)|^2$ as a double sum. Diagonal terms (where $n=m$) yield main terms, off-diagonal ($n\neq m$) govern error terms and are controlled using oscillatory integral bounds.
• Oscillatory Integrals and Integration by Parts: Bounds for integrals of the type $\int A(x) e(B(x)) dx$ are obtained by repeated integration by parts, contingent on $A$'s smoothness and bounds for $B'(x)$ (see Lemma 2.1 in (Ernvall-Hytönen, 2010)).
• Trigonometric Identities: To handle differences of oscillatory terms, identities such as $$\cos\theta_1-\cos\theta_2 = -2\sin\left(\frac{\theta_1+\theta_2}{2}\right)\sin\left(\frac{\theta_1-\theta_2}{2}\right)$$ are used to extract secondary cancellation structure.

The result for short exponential sums is the mean square bound

$$\int_{M}^{M+A} \left| \sum_{n \in [x, x+V(x)]} a(n) e(hn/k) \right|^2 x^{1/2} w(x) dx \ll k A M^{1/2}$$

demonstrating on average the conjectural "square-root cancellation" (Ernvall-Hytönen, 2010).

3. Riesz Sums in Quadratic and Resonant Settings

In the context of quadratic progressions,

$$S(d, Y) = \sum_n a(n^2 + d) W\left( \frac{n^2+d}{Y} \right)$$

are "Riesz sums" in the variable $n$ over quadratic shifts (Templier et al., 2011, Kuan et al., 2023). Here, cancellation properties and main term formation depend on the arithmetic nature of the shift $d$, the smoothness of $W$, and the automorphic/nature of $f$.

Templier–Tsimerman (Templier et al., 2011) demonstrate that for such non-split sums (over quadratic sequences rather than linear), the main term is present only if the underlying form is dihedral, and the error term is uniform in both $d$ and $Y$, critically reflecting the depth of uniformity possible in Riesz sum analysis.

For "resonance sums" (Riesz sums with oscillatory twists) of the form

$$S_X(f,g;\alpha,\beta) = \sum_{X \le n \le 2X} \lambda_f(n)\lambda_g(n) e(\alpha n^\beta)\phi\left(\frac{n}{X}\right),$$

breakthroughs in square-root cancellation on average (over weights or over families of forms) have recently been achieved by combining Petersson's trace formula, Poisson summation, and stationary phase analysis (Gillespie et al., 2022). For almost all pairs $(f, g)$, strong cancellation is established beyond the "resonance barrier," with square-root behavior for many ranges of $\beta$.

4. Error Terms, Asymptotics, and Uniformity

A key outcome of modern Riesz sum analysis is precise error terms and uniformity in relevant parameters. For example:

• The main term in non-split quadratic sums arises only for special arithmetic ("dihedral") automorphic forms and vanishes in most cases, reflecting deep symmetry (Templier et al., 2011).
• Error terms depend both on Selberg eigenvalue conjecture exponents ($\delta$) and on bounds for half-integral weight Fourier coefficients ($\theta$), leading to error terms like $O_\epsilon(d^{-1/2+\theta} Y^{\beta-1/2+\epsilon})$ (Templier et al., 2011).

For the sharp cut-off sum $\sum_{n^2+h<X^2}A(n^2+h)$, recent results achieve

$$\sum_{n^2+h \le X^2} A(n^2+h) = c_{f,h} X + O_{f,h,\epsilon}(X^{3/4+\epsilon})$$

which improves earlier work and directly informs Riesz means by allowing transfer of error bounds from the sharp-cutoff case to weighted means (Kuan et al., 2023).

5. Spectral and Representation-Theoretic Aspects

The analysis of Riesz sums also fundamentally draws on spectral expansion, trace formula, and representation theory:

• Kuznetsov and Petersson Formulas: These are central when associating moments of Riesz sums (or zero statistics, as in low-lying zero densities) to explicit trace formulas and their harmonic weights (Hulse et al., 2015, Amersi et al., 2011).
• Spectral/Arithemtic Duality: For higher moments or products of Fourier coefficients (e.g., quartic spectral sums in weight $1/2$), Bruggeman–Kuznetsov-type formulas expressing spectral sums in terms of generalized class numbers enable explicit spectral-arithmetic duality for Riesz-type aggregates (Biró, 3 Jul 2025).
• Voronoï Summation: The Voronoï formula connects Riesz (or twisted) sums to dual sums involving Bessel transforms, enabling nontrivial bounds and uniformity in level, weight, and modulus (Assing et al., 2019).

6. Broader Implications and Connections

Riesz sums attached to cusp forms serve as analytic benchmarks for several deep phenomena:

• Subconvexity and Mean Values of $L$-functions: Riesz mean estimates tie directly to moments of $L$-functions and subconvexity values via mean-square and shifted convolution considerations.
• Zero Density and Nonvanishing: The size of Riesz sums, especially under twists or over short intervals, is sensitive to the density and distribution of zeros of relevant automorphic $L$-functions (Frechette et al., 2023).
• Equidistribution and Quantum Ergodicity: Spectral Riesz sums enter in quantum unique ergodicity and prime geodesic theorems by tracking the fluctuations of spectral measures and their connections to modular symbols (Kaneko, 2019).

7. Key Formulas

Setting	Riesz-Type Sum	Error/Main Term
Short Exponential Sum	$\int_{M}^{M+A} |S(x)|^2 x^{1/2} w(x) dx$	$\ll k A M^{1/2}$
Quadratic Progression	$\sum_n a(n^2+d) W((n^2+d)/Y)$	$I(W) M_{f,d} Y^\beta + O_\epsilon(\cdots)$
Partial Sum	$S(x, f) = \sum_{n < x} \lambda_f(n)$	$o(x \log x)$ (under zero-density assumptions)
Resonance Sums	$S_X(f,g;\alpha,\beta)$	$\ll X^{1/2+\epsilon}$ (for most $f,g$ with $\beta$ large enough)
Convolution Second Moment	$\sum_{n\le X} |S_f(n)|^2$	$\sim X^{k-1+1/2}$ up to $X^\theta$ error (on average)

Riesz sums thus yield sharp mean value results, uniform error control, and invite a detailed analysis of both analytic and arithmetic inputs, with rich interplays between spectral expansions, oscillatory integral analysis, and automorphic representation theory.