[Paper Review] On monotonicity of certain weighted summatory functions associated with L-functions

Masatoshi Suzuki|arXiv (Cornell University)|Apr 9, 2012

Analytic Number Theory Research11 references3 citations

TL;DR

This paper establishes that the monotonicity of weighted summatory functions $ h_{f, heta}^{ angle k angle}(x) $, derived from $ L $-functions $ L(f,s) $ with specific weight functions depending on parameters $ \omega \in (0,1/2) $ and $ k \in \mathbb{N} $, is equivalent to the Generalized Riemann Hypothesis (GRH) for those $ L $-functions. The proof uses complex analysis, Mellin transforms, and asymptotic analysis of integrals involving $ \Theta_{f,\omega}(s) $, showing that the sign of $ h_{f,\omega}^{ angle k\rangle}(x) $ stabilizes for large $ x $, which characterizes GRH.

ABSTRACT

We state some sufficient or equivalent conditions to GRH of general L-functions in terms of monotonicity of certain weighted summatory functions.

Motivation & Objective

• To establish a sufficient and necessary condition for the Generalized Riemann Hypothesis (GRH) in terms of the monotonicity of weighted summatory functions associated with $ L $-functions.
• To generalize classical results such as Pólya’s and Chebyshev’s conjectures by introducing a family of weight functions parameterized by $ \omega \in (0,1/2) $ and $ k \in \mathbb{N} $.
• To show that the sign of the weighted summatory function $ h_{f,\omega}^{ angle k\rangle}(x) $ stabilizes for large $ x $, which is equivalent to the monotonicity of its integral.
• To extend known equivalence results (e.g., for $ L(s,\chi_4) $) to general $ L $-functions via a unified analytic framework.

Proposed method

• The paper defines a weighted summatory function $ h_{f,\omega}^{ angle k\rangle}(x) = \sum_{n=1}^\infty c(n) g_\omega\left(\frac{n}{x}\right) $, where $ g_\omega $ is a weight function involving incomplete beta functions and gamma functions.
• It uses the Mellin transform of the weight function to relate $ h_{f,\omega}^{ angle k\rangle}(x) $ to the $ L $-function $ L(f,s) $, via the generating function $ \Theta_{f,\omega}(s) $.
• Complex analysis techniques are applied, including contour integration over rectangles in the critical strip, with the use of the functional equation and analytic continuation of $ \Theta_{f,\omega}(s) $.
• The asymptotic behavior of the integrals is analyzed using the Riemann-Lebesgue lemma and Stirling’s formula, showing that $ h_{f,\omega}^{ angle k\rangle}(x) \sim \varepsilon(f)(\log x)^{k-1} $ as $ x \to \infty $.
• The $ L^2 $-integrability of the remainder term $ R_{f,\omega}(x) $ is used to prove that $ (\Theta_{f,\omega}(s) - \varepsilon(f))/(s - 1/2) $ has no poles in $ \Re(s) > 1/2 $, which implies GRH.
• The proof relies on the Ramanujan-Petersson conjecture for $ \lambda_f(n) $ and $ \mu_f(n) $, ensuring polynomial growth bounds for the coefficients.

Experimental results

Research questions

• RQ1 Is the monotonicity of the integral $ \int_1^x h_{f,\omega}^{\rangle k\rangle}(t)\,dt $ for large $ x $ equivalent to the GRH for $ L(f,s) $?
• RQ2 Can the classical equivalence results (e.g., for $ L(s,\chi_4) $) be generalized to arbitrary $ L $-functions using a one-parameter family of weight functions?
• RQ3 Does the sign of $ h_{f,\omega}^{\rangle k\rangle}(x) $ stabilize for large $ x $, and if so, what does this imply about the location of non-trivial zeros of $ L(f,s) $?
• RQ4 How does the choice of weight function $ g_\omega $, involving incomplete beta functions, affect the asymptotic behavior of the summatory function?

Key findings

• For any fixed $ k \geq 2 $, the monotonicity of $ \int_1^x h_{f,\omega}^{\rangle k\rangle}(t)\,dt $ for large $ x $ is equivalent to the GRH for $ L(f,s) $.
• The asymptotic expansion of $ h_{f,\omega}^{\rangle k\rangle}(x) $ is $ \varepsilon(f)(\log x)^{k-1} \left(1 + O((\log x)^{-1})\right) $, showing that the function does not change sign for large $ x $.
• For $ k = 1 $, the function $ h_{f,\omega}^{\rangle 1\rangle}(x) = \varepsilon(f) + O(x^{-B}) $ for some $ B > 0 $, implying sign stability and equivalence to GRH.
• The remainder term $ R_{f,\omega}(x) $ is in $ L^2((1,\infty), x^{-1}dx) $, which ensures that $ (\Theta_{f,\omega}(s) - \varepsilon(f))/(s - 1/2) $ has no poles in $ \Re(s) > 1/2 $, a key step in proving GRH equivalence.
• The equivalence holds uniformly for all $ \omega \in (0,1/2) $, generalizing earlier results that were restricted to specific weight functions like $ \exp(-y) $ or $ \exp(-y^\alpha) $.