Sums of Kloosterman sums formed with modular symbols

Abstract: We study sums of Kloosterman sums formed with a modular symbol. Employing Tauberian methods, we first give an estimate for a (Riesz) sum of Ramanujan sums formed with a modular symbol. We further define a zeta function that is analogous to the Selberg zeta function, establish its continuation to $\Re(s)>1/2$, give estimates for its growth and use this to prove a cancellation statement for sums of these twisted Kloosterman sums. We explain the connection of this construction to the eigenvalue 1/4 problem and formulate an analogue of Linnik's conjecture. Finally, we present numerical evidence that there is cancellation and also that the Kloosterman sums with a modular symbol are not correlated with classical Kloosterman sums.

• The paper establishes meromorphic continuation and growth bounds for a Selberg-type zeta function formed with modular symbols.
• It applies extended Tauberian methods to derive cancellation estimates for twisted Ramanujan sums and asymptotic error terms of O(X^(1/2+ε)).
• The study confirms statistical independence between classical and modular-symbol-twisted Kloosterman sums, showing no significant correlation as the modulus increases.

Sums of Kloosterman Sums Formed with Modular Symbols: A Technical Analysis

Introduction and Motivations

This work focuses on sums of Kloosterman sums that are "twisted" by modular symbols—expressions at the intersection of analytic number theory and the spectral theory of automorphic forms. The authors generalize classical Kloosterman sums by weighting their terms with modular symbols associated to holomorphic cusp forms. Specifically, the sum

$$S^*(m,n;c) = \sum_{\substack{\gamma\in \Gamma_\infty \backslash \Gamma_0(N)/\Gamma_\infty \ c_\gamma=c}} \langle f, \gamma \rangle \exp\left(2\pi i (m a_\gamma + n d_\gamma)/c_\gamma\right)$$

is examined, where the modular symbol $\langle f, \gamma \rangle$ replaces the standard coefficient.

This setting connects to deep conjectures in the theory of automorphic representations and the spectral theory of the Laplacian, notably the 1/4-eigenvalue conjecture for Maass forms and analogues of the Linnik and Selberg conjectures regarding cancellation in sums of Kloosterman sums. The analysis is inherently technical, combining Tauberian theorems, the analytic theory of Selberg-type zeta functions, and perturbative spectral methods.

Main Analytical Tools and Results

The authors extend Tauberian techniques to sums of Kloosterman sums weighted by modular symbols, firstly for the simpler case corresponding to (twisted) Ramanujan sums, then for the genuinely Kloosterman context. They construct and study a modified "Selberg zeta function" $Z^*_{m,n}(s)$ attached to these twisted sums, establishing its meromorphic continuation to $Re(s)>1/2$. They also bound its growth and analyze its singularities.

Estimate for Twisted Ramanujan Sums

Theorem~\ref{main} (paraphrased) uses Tauberian methods to show that for $n \neq 0$ and $N=p$ prime,

$$\sum_{c < (\pi X|n|)^{1/2}/p} S^*(n,0; pc) \psi_c(n, X) = MT + O(X^{1/2 + \epsilon})$$

with explicit main term $MT$ and weights $\psi_c$, of order $X^{1 + \epsilon}$ as $\langle f, \gamma \rangle$0.

This result provides a "smoothed sum" version of cancellation for these twisted sums, analogous to classical results on Kloosterman sums but with additional technical complications from infinite poles on the critical line and the non-unit magnitude of modular symbols.

Selberg-type Zeta Function and Analytic Continuation

They define a Dirichlet series

$\langle f, \gamma \rangle$1

and show, using Sobolev space methods and pseudo-Laplacians à la Phillips-Sarnak/Petridis, that:

• $\langle f, \gamma \rangle$2 continues meromorphically to $\langle f, \gamma \rangle$3,
• the only possible poles in $\langle f, \gamma \rangle$4 are dictated by exceptional Laplacian eigenvalues,
• away from poles, $\langle f, \gamma \rangle$5 has controlled polynomial growth in $\langle f, \gamma \rangle$6.

This involves an elaborate reduction to a perturbation of Eisenstein series, controlling the analytic properties of Poincaré-type and Niebur series with modular symbol twists.

Asymptotic Statement for Sums

The analogue of Goldfeld-Sarnak's result ([GS]) is obtained: For the average of the twisted Kloosterman sum,

$\langle f, \gamma \rangle$7

where $\langle f, \gamma \rangle$8, $\langle f, \gamma \rangle$9 relate to possible exceptional eigenvalues, and $Z^*_{m,n}(s)$0 is the maximal abscissa of convergence, i.e., the maximal possible growth rate of $Z^*_{m,n}(s)$1.

Numerical Results, Cancellation, and Non-correlation

A critical aspect is the extensive computational investigation. The authors verify, across varying levels and parameters, that:

• The conjectured $Z^*_{m,n}(s)$2 cancellation (Linnik–Selberg type) in the twisted sums is broadly observed,
• There is no evidence for correlation between classical and modular-symbol-twisted Kloosterman sums; their joint sign and Pearson correlations trend toward $Z^*_{m,n}(s)$3 and $Z^*_{m,n}(s)$4 respectively in the large $Z^*_{m,n}(s)$5 limit.

Figure 1: Empirical correlation $Z^*_{m,n}(s)$6 between signs of $Z^*_{m,n}(s)$7 and $Z^*_{m,n}(s)$8, showing no significant correlation as $Z^*_{m,n}(s)$9 increases.

Figure 2: Pearson correlation $Re(s)>1/2$0 between classical and modular-symbol-twisted Kloosterman sums across dyadic windows; rapid decay indicates statistical independence.

This lack of correlation is formalized as Conjecture~\ref{no-correlation}.

Tauberian Theorems and Remainder Term Analysis

With technical computations, the authors analyze the remainder terms in the Tauberian summation for $Re(s)>1/2$1. The experimental fits with the predicted boundary integrals (Eisenstein term) and Maass cusp form contributions demonstrate that:

• The expected order for the error is $Re(s)>1/2$2,
• Any putative exceptional eigenvalue would result in a detectable power saving/worsening, which is not observed.

Figure 3: The relative error $Re(s)>1/2$3 in log-log scale for various $Re(s)>1/2$4, $Re(s)>1/2$5, $Re(s)>1/2$6; lines of slope $Re(s)>1/2$7 indicate consistency with predicted theory.

Figure 4: Decomposition of $Re(s)>1/2$8 showing oscillatory (Maass) and monotone (Eisenstein) components.

Figure 5: Fine structure of remainder: separation between Eisenstein and Maass layers globally matches analytic predictions.

The explicit connection between the lack of exceptional spectrum and observed cancellation in these sums is made rigorous; conjectures regarding the non-vanishing of the associated residues (cf. Conjecture~\ref{conj-ram-sums}) would, if true, imply the absence of exceptional eigenvalues.

Implications and Future Directions

The analysis offers new Tauberian theorems and analytic techniques for sums incorporating both automorphic and cohomological (modular symbol) weights. The computational results suggest powerful random-matrix–type cancellation in these "higher rank" summation scenarios, mimicking the theoretical predictions of spectral theory in arithmetic.

From a broader perspective, the methods provide a blueprint for analogous studies involving other "twisted" automorphic Dirichlet series, such as those weighted by periods, additive twists, or higher-order modular symbols. The robust toolkit combining analytic continuation, functional equations, and perturbation theory of operators will have further impact on the study of special values of $Re(s)>1/2$9-functions and trace formulae.

Conclusion

This paper provides a comprehensive analytic and experimental study of modular-symbol-weighted Kloosterman sums. Through a combination of Tauberian, spectral, and computational techniques, it:

• Establishes polynomial-type cancellation in averages of these sums,
• Demonstrates the analytic continuation of the associated Selberg-type zeta function,
• Quantitatively relates the asymptotics to the presence of exceptional Laplacian eigenvalues,
• Empirically confirms cancellation and statistical independence from classical Kloosterman sums,
• Suggests a strategy for further exclusion of exceptional spectrum via refined residue analysis.

The results reinforce the rich interplay between spectral theory, analytic number theory, and arithmetic geometry, illuminating open paths for further study in automorphic representation theory and $n \neq 0$0-function theory.

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References

• Original preprint: "Sums of Kloosterman sums formed with modular symbols" (2607.10786).
• Classical: [GS] D. Goldfeld and P. Sarnak, "Sums of Kloosterman sums," Invent. Math. 71, 243–250, 1983.