[Paper Review] Oscillations of Hecke Eigenvalues at Primes

Liangyi Zhao|ArXiv.org|Aug 7, 2005

Analytic Number Theory Research14 references3 citations

TL;DR

This paper establishes a non-trivial hybrid bound for exponential sums involving Hecke eigenvalues twisted by square-root amplitude at prime arguments. Using Vinogradov's method, Perron's formula, and mean value theorems for automorphic L-functions, the author proves that the sum $ S(N) = \sum_{n \leq N} \lambda(n) \Lambda(n) e(\alpha \sqrt{n}) $ is bounded by $ O(N^{5/6} [\log(3N)]^{21}) $, improving upon the classical Vinogradov-type bound and approaching the conjectured $ N^{3/4} $-level under strong hypotheses.

ABSTRACT

In this paper, we are interested in exploring the cancellation of Hecke eigenvalues twisted with an exponential sums whose amplitude is $\sqrt{n}$ at prime arguments.

Motivation & Objective

• To estimate the exponential sum $ S(N) = \sum_{n \leq N} \lambda(n) \Lambda(n) e(\alpha \sqrt{n}) $, where $ \lambda(n) $ are Hecke eigenvalues of a cusp form and $ \Lambda(n) $ is the von Mangoldt function.
• To investigate the oscillatory behavior of Hecke eigenvalues when twisted with exponential sums of square-root amplitude at prime arguments.
• To improve upon classical Vinogradov-type bounds for such sums, approaching the conjectured $ N^{3/4} $-level under strong hypotheses.
• To demonstrate that the current technology, when combined with mean value theorems for automorphic L-functions, yields a non-trivial hybrid bound of $ O(N^{5/6} [\log(3N)]^{21}) $.

Proposed method

• Application of Vinogradov's method to estimate exponential sums with square-root amplitude, decomposing the sum into dyadic intervals.
• Use of Perron's formula to express partial sums as complex integrals, followed by shifting the line of integration to the critical line $ \text{Re}(s) = 1/2 $.
• Employment of the multiplicative property of Hecke eigenvalues (via Möbius inversion) to decompose bilinear forms in the sum.
• Application of the large sieve and mean value theorems for automorphic L-functions to bound the integrals involving $ L(1/2 + it, f) $ and its derivative.
• Estimation of truncated Möbius convolutions involving $ \Lambda(n) $ and $ \nu(d) $, with careful dyadic decomposition and partial summation.
• Use of the Ramanujan conjecture and Deligne's bound $ |\lambda(n)| \ll n^{\epsilon} $ to control the size of Hecke eigenvalues in the sums.

Experimental results

Research questions

• RQ1 What is the best possible cancellation in exponential sums of the form $ \sum_{n \leq N} \lambda(n) \Lambda(n) e(\alpha \sqrt{n}) $, where $ \lambda(n) $ are Hecke eigenvalues of a cusp form?
• RQ2 Can the classical Vinogradov bound $ O(N^{7/8 + \epsilon}) $ be improved for this hybrid sum involving both arithmetic functions and oscillatory exponential terms?
• RQ3 What role do mean value theorems for automorphic L-functions play in bounding such bilinear forms with Hecke eigenvalues?
• RQ4 Why does the current method fall short of achieving the conjectured $ N^{3/4} $-level, and what are the key obstructions?
• RQ5 Under what assumptions (e.g., Lindelöf hypothesis or Montgomery’s conjecture) might the bound be improved to $ O(N^{3/4 + \epsilon}) $?

Key findings

• The sum $ S(N) = \sum_{n \leq N} \lambda(n) \Lambda(n) e(\alpha \sqrt{n}) $ is bounded by $ O(N^{5/6} [\log(3N)]^{21}) $, where the implied constant depends on $ \alpha $ and the cusp form $ f(z) $.
• The bound is achieved by decomposing the sum into Type I and Type II sums, applying Perron's formula, and using mean value theorems for $ L(1/2 + it, f) $.
• The Type I sum is bounded by $ O(N^{3/4 + \epsilon}) $, and the Type II sum is controlled via bilinear form estimates and the multiplicative structure of $ \lambda(n) $.
• The result is conditional on the strength of current mean value theorems; stronger bounds on $ L $-functions would yield the conjectured $ N^{3/4} $-level.
• The method is robust and extends to cusp forms for congruence subgroups, though the resulting bound weakens with increasing level.
• The paper confirms that the $ N^{3/4} $-level is not currently accessible with existing tools, and the main obstacle lies in estimating bilinear forms more efficiently.