Core Concepts

This paper establishes an unconditional asymptotic formula for the moments of S(t), a function related to the argument of L-functions associated with even Hecke-Maass cusp forms for SL2(Z), and proves the distribution of these values approaches a normal distribution as the parameter T tends to infinity.

Abstract

**Bibliographic Information:**Sun, Q., & Wang, H. (2024). On an unconditional spectral analog of Selberg’s result on S(t).*arXiv preprint arXiv:2410.03473*.**Research Objective:**This research paper aims to establish an unconditional asymptotic formula for the moments of S(t), a function related to the argument of L-functions associated with even Hecke-Maass cusp forms for SL2(Z). This extends previous work that relied on the Generalized Riemann Hypothesis (GRH).**Methodology:**The authors utilize the Kuznetsov trace formula over even Hecke-Maass forms, along with a weighted version of the zero-density estimate in the spectral aspect for the associated L-functions. They derive an approximation for S(t) and analyze its behavior as the parameter T tends to infinity.**Key Findings:**The paper successfully proves an unconditional asymptotic formula for the moments of S(t) for even Hecke-Maass cusp forms. Additionally, they demonstrate that the distribution of the ratios S(t)/√(log log T) converges in distribution to a standard normal distribution as T approaches infinity.**Main Conclusions:**This research provides a significant advancement in understanding the behavior of S(t) for L-functions associated with even Hecke-Maass cusp forms, particularly by removing the dependence on the GRH. The established asymptotic formula and the convergence to a normal distribution offer valuable insights into the statistical properties of these L-functions.**Significance:**This work contributes significantly to the field of analytic number theory, specifically to the study of L-functions and their spectral properties. The removal of the GRH assumption represents a substantial step forward in this area.**Limitations and Future Research:**The paper focuses specifically on even Hecke-Maass cusp forms for SL2(Z). Further research could explore extending these results to other types of L-functions or more general groups. Additionally, investigating the implications of these findings for related problems in number theory would be a promising avenue for future work.

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by Qingfeng Sun... at **arxiv.org** 10-07-2024

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Extending the results of this paper to L-functions associated with other automorphic forms, such as Maass forms with higher level or forms on higher rank groups, is a natural and challenging direction for further research. Here's a breakdown of the potential challenges and approaches:
Challenges:
Spectral Theory: The Kuznetsov trace formula, a crucial tool in this paper, becomes more intricate for higher level groups. Generalizations exist, but they often involve more complicated terms and require a deeper understanding of the underlying spectral theory.
Ramanujan-Petersson Conjecture: The current proof relies on the best known bounds towards the Ramanujan-Petersson conjecture for the specific type of Maass form considered. For other automorphic forms, especially on higher rank groups, the analogous conjectures might be even deeper and lack comparable bounds. This would necessitate developing new techniques to handle the sums over Fourier coefficients.
Zero-Density Estimates: The weighted zero-density estimate used here is a powerful tool. Obtaining analogous estimates for L-functions on higher rank groups or with higher level is a non-trivial task and often an active area of research itself.
Approaches and Potential:
Congruence Subgroups: One could start by considering Maass forms for congruence subgroups of SL2(Z). The Kuznetsov trace formula for these groups is well-studied, and some progress might be possible by adapting the existing methods.
Rankin-Selberg L-functions: For higher rank groups, one could investigate Rankin-Selberg L-functions associated with lower-rank groups. These L-functions often have better analytic properties, and existing techniques might be more readily applicable.
Families of L-functions: Instead of focusing on individual L-functions, one could study families of L-functions associated with varying automorphic forms. This approach has been fruitful in other contexts and might offer a way to circumvent some of the technical difficulties associated with individual L-functions.
In summary, extending these results to a broader class of L-functions is a significant undertaking that would require overcoming several technical hurdles. However, the potential rewards are high, as it would lead to a deeper understanding of the analytic behavior of L-functions and their connections to the underlying automorphic forms.

Yes, the methods employed in this paper, which blend techniques from analytic number theory, automorphic forms, and probability theory, hold promise for studying the distribution of values of other arithmetic functions connected to L-functions. Here are some potential avenues:
Moments of L-functions: The core strategy of analyzing moments of the argument of L-functions can be adapted to investigate moments of other L-function-related quantities. For instance, one could study moments of the logarithmic derivative of L-functions near the critical line, which are known to encode information about the distribution of zeros.
Mollifiers: The use of smooth weight functions (mollifiers) to dampen the oscillations of L-functions is a versatile technique. It could be applied to study the distribution of values of L-functions in families or to investigate the frequency of large values.
Central Values of L-functions: The distribution of central values of L-functions (values at s=1/2) is a central topic in number theory. The methods in this paper, particularly the use of the Kuznetsov trace formula and zero-density estimates, could potentially be adapted to study the distribution of these central values in various families of L-functions.
Special Values of L-functions: Beyond central values, special values of L-functions at other points within the critical strip are also of great interest. The techniques employed here could be modified to investigate the distribution of these special values and their connections to arithmetic invariants.
The key takeaway is that the interplay between analytic tools for handling L-functions and probabilistic methods for studying distributions provides a powerful framework. This framework can be extended and modified to explore the statistical behavior of a wide range of arithmetic functions intertwined with L-functions.

The convergence of S(t)/√(log log T) to a normal distribution, as shown in this paper and related works, provides profound insights into the statistical nature of the zeros of L-functions. Here's a breakdown of the implications:
Repulsion of Zeros: The normal distribution result suggests a form of "repulsion" between zeros of the L-function. If zeros were clustered together or spaced randomly, we would expect different limiting distributions for S(t)/√(log log T). The normal distribution indicates a more even spacing of zeros on average.
Connection to Random Matrix Theory: The appearance of the normal distribution aligns with predictions from Random Matrix Theory (RMT). RMT postulates that the statistical properties of zeros of L-functions, in certain families, should resemble the eigenvalues of random matrices. The normal distribution arising in both contexts strengthens the link between L-functions and RMT.
Support for the GRH: While not a proof, the convergence to a normal distribution is consistent with the Generalized Riemann Hypothesis (GRH). The GRH dictates a very precise distribution of zeros, and the observed normal distribution behavior aligns with what one would expect under GRH.
Understanding the Error Term in the Prime Number Theorem: The function S(t) is intimately connected to the error term in the Prime Number Theorem. The normal distribution result provides insights into the fluctuating behavior of this error term and sets limitations on how large (or small) it can be.
In essence, the convergence of S(t)/√(log log T) to a normal distribution reveals a hidden order in the seemingly chaotic distribution of zeros of L-functions. This order has profound connections to other areas of mathematics, particularly random matrix theory, and provides compelling evidence in favor of the GRH and a deeper understanding of prime numbers.

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