Shifted Convolution Sums in Number Theory

Updated 20 October 2025

Shifted convolution sums are defined as arithmetic sums of the form ∑ a(n) b(n+h) that capture correlations between functions like Hecke eigenvalues and divisor functions.
They exhibit a critical transition region where the asymptotic behavior shifts from pointwise estimates to main term dominance as the ratio Y²/X varies.
Spectral methods, including expansions into Eisenstein series and Maass forms, alongside multiple Dirichlet series, are key to deriving precise asymptotic formulas and error estimates.

A shifted convolution sum is an arithmetic sum of the form $\sum_{n} a(n)\, b(n+h)$ , where $a(n)$ and $b(n)$ are arithmetic or automorphic functions—most notably Hecke eigenvalues or divisor functions—and $h$ is a nonzero integer "shift." Such sums, and their averages over the shift, are central in analytic number theory. They encode correlations between arithmetic functions across different arguments, are intimately connected with moments and values of $L$ -functions, and play a key role in subconvexity estimates, quantum unique ergodicity, and the spectral theory of automorphic forms.

1. Fundamental Definition and Classical Context

The classical instance of a shifted convolution sum is

$S(h; N) = \sum_{n\leq N} a(n)\, b(n+h),$

where $a(n)$ , $b(n)$ could be, for example, divisor functions $d(n)$ , Fourier coefficients of modular forms $\lambda_f(n)$ , or general multiplicative functions. This structure is general: in the automorphic context for a holomorphic cusp form $f$ for $SL_2(\mathbb{Z})$ , the basic form is

$\sum_n \lambda_f(n)\lambda_f(n+h).$

These sums, in contrast to self-correlation ( $h=0$ ) sums often fully understood via Rankin–Selberg theory, are sensitive to deep arithmetic and cancelation phenomena when $h\neq 0$ .

Historically, they arise in the paper of the additive divisor problem (Ingham, Estermann) and appear naturally in the context of subconvexity for $L$ -functions, through the spectral theory of automorphic forms, and in problems concerning equidistribution, such as quantum unique ergodicity (Petrow, 2011, Nordentoft et al., 2021).

2. Asymptotics, Averaging, and the Transition Phenomenon

A central theme is the derivation of asymptotic formulas, especially after averaging over $h$ or $n$ and introducing smoothing weights. The work (Petrow, 2011) develops a general framework for such averaged shifted convolution sums: $S_f(\psi, Y) = \sum_{h\approx Y} \sum_{n\approx X} \lambda_f(n)\lambda_f(n+h)\psi\left(\frac{n}{X}\right),$ where $\psi$ is a smooth cutoff. The paper proves an explicit asymptotic expansion: $S_f(\psi, Y) = \frac{1}{(4\pi)^k}\int_{0}^{\infty} \left[c_f(4\pi yY^2) - \frac{\Gamma(k) L(1,\mathrm{sym}^2 f)}{2 \zeta(2)}\right] \frac{\psi(y)}{y^2} dy + O_k\left( Y^{\frac{1}{3}(1+\theta)} \int_0^\infty \frac{|\psi(y)|}{y^{3/2}} dy \right),$ with $c_f$ an explicit function defined by

$c_f(\alpha) = \frac{\pi^{3/2}}{2}\, \alpha \sum_{n=1}^{\infty} \lambda_f(n)^2 W_k(\pi^2 n \alpha),$

where $W_k$ involves integrals of Gamma functions.

A fundamental finding is the existence of a transition region: the asymptotic behavior of $S_f(\psi,Y)$ is controlled by the ratio $Y^2/X$ between the square of the shift range and the length of the $n$ -sum. For $Y^2/X \ll 1$ , pointwise bounds dominate, while for $Y^2/X \gg 1$ , the main term is governed by Rankin–Selberg theory. In the critical "transition" regime $Y^2/X \asymp \mathrm{const}$ , the main term undergoes a continuous change, captured by the function $c_f(\alpha)$ interpolating between the two extremes. The phenomenon connects to the analytic behavior of multiple Dirichlet series and Eisenstein series, as found in similar contexts by Conrey–Farmer–Soundararajan (Petrow, 2011).

3. Spectral and Analytic Techniques

The determination of the main terms and error terms in shifted convolution sums leverages spectral decompositions and analytic number theory techniques:

Spectral Theory: The expansion of incomplete Poincaré series associated to the shift using the spectral decomposition into Eisenstein series and Maass forms is central (Petrow, 2011). The Eisenstein series part encodes the main term; Maass forms contribute to the error term. This spectral machinery brings in explicit integral transforms, Dirichlet series, and Bessel function analysis, reflecting the automorphic origin of the problem.
Multiple Dirichlet Series: The analytic continuation and the structure of the associated series explain the transition region: poles of the series collide as the length scales become commensurate, producing the observed "blow-up" and smoothing of the main term.

For more general settings involving higher rank or different families, methods such as the circle method (exploiting modular invariance and additive characters), delta-method decompositions, and delicate bounding of character sums (as in the factorization and use of trace function estimates) are used (Munshi, 2012, Xi, 2017).

Shifted convolution sums appear in several guises:

Jacobi Symbol and Character Averages: Earlier work by Conrey–Farmer–Soundararajan studies shifted character sums and exhibits similar transition phenomena, though with different analytic objects (Jacobi symbol, Dirichlet L-functions). Both their results and the results on Hecke eigenvalue correlations exhibit transition regions governed by the length scale ratio, arising via the analytic structure of multiple Dirichlet series.
Automorphic and Quantum Chaos: The local to global spectral correspondence, especially in the context of automorphic forms for $\mathrm{SL}_2(\mathbb{Z})$ , means that shifted convolution sums control quantum unique ergodicity questions and the distribution of mass for eigenfunctions.

There are comparisons in smoothness of the transition: while the main term in the Jacobi symbol case may have non-differentiable singularities, for the cusp form case (after smoothing) the main term function $c_f$ is $C^\infty$ on $(0, \infty)$ , but with explicit limiting behavior as seen above.

5. Applications and Broader Impact

Asymptotic results for shifted convolution sums, especially with controlled error terms, have ramifications across analytic number theory:

Subconvexity for $L$ -functions: Sharp estimates for shifted sums feed into mean value theorems and subconvexity bounds for automorphic $L$ -functions as these sums appear in off-diagonal terms of moments, and approximate functional equations.
Quantum Unique Ergodicity: The best known results for mass equidistribution of cusp form eigenfunctions on arithmetic surfaces require control over shifted convolution sums (Petrow, 2011).
Analytic Structure of Automorphic Forms: The interplay between Eisenstein series, spectral expansions, and shifted convolution sums is emblematic of the deep analytic and algebraic structure of the trace formula and spectral theory in automorphic forms, including the behavior of Fourier coefficients and their correlations.

6. Technical and Methodological Innovations

Notable methodological points include:

Explicit exploitation of incomplete Poincaré series and their expansions.
Precise quantification and smoothing of the length parameters, enabling the analysis of transition behavior.
Detailed computation and control of error terms, using bounds for Maass forms and estimates such as $Y^{(1/3)(1+\theta)}$ .
Introduction and analysis of smooth cutoff functions and their effect on the analytic continuation of multiple Dirichlet series and Eisenstein series.
Drawing connections to multiple Dirichlet series analysis, as appears at the interface of automorphic forms, special functions, and analytic number theory.

The transition phenomenon and the spectral methods developed for shifted convolution sums have inspired several further developments:

The approach is expected to extend to other problems where dual length scales interact—such as multiple shifted convolution sums, higher rank analogues, and sums over different families (e.g., $GL(3) \times GL(2)$ coefficients).
The explicit relation to multiple Dirichlet series and zeta functions highlights potential for interaction with the analytic theory of Euler products, moments of $L$ -functions, and the theory of singular divisors.
The identification of transition regions motivates future research into the interplay of scaling regimes and spectral expansions, possibly leading to new phenomena where analytic and arithmetic boundaries coincide.

Shifted convolution sums remain a central object of paper, both as a deep analytic problem in their own right and as a tool for advancing major conjectures in number theory. The phenomenon of transition mean values as developed in (Petrow, 2011) provides a paradigm for understanding the subtle ways in which multiple length scales and automorphic analysis interact.