--- title: Riesz Sum Analysis for Cusp Forms url: https://www.emergentmind.com/topics/riesz-sum-attached-to-a-cusp-form type: topic --- # Riesz Sum Analysis for Cusp Forms 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 study 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 $k$ [1005.3690]. - **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 [1106.1139]. - **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 [2209.03856]. 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 $e^{-s\sqrt{n}}$ [2311.00006]). ## 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 [1005.3690]. 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 [1005.3690]). - **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" [1005.3690]. ## 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 [1106.1139, 2301.11901]. 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 [1106.1139] 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 [2209.03856]. 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 [1106.1139]. - 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 $O_\epsilon(d^{-1/2+\theta} Y^{\beta-1/2+\epsilon})$ [1106.1139]. For the sharp cut-off sum $\sum_{n^2+h