[Paper Review] Negative values of truncations to L(1

Andrew Granville, K. Soundararajan|arXiv (Cornell University)|Aug 18, 2005

Graph theory and applications5 references3 citations

TL;DR

This paper establishes upper and lower bounds for the minimum of the truncated sum ∑_{n≤x} χ(n)/n over real-valued Dirichlet characters χ, and extends the analysis to completely multiplicative, real-valued functions f with |f(n)| ≤ 1. The key result is that when the domain is expanded to all multiplicative, real-valued functions with |f(n)| ≤ 1, the infimum of the sum approaches approximately -0.4553 as x → ∞, with a complete classification of the optimal functions achieving this bound.

ABSTRACT

For flxed large x we give upper and lower bounds for the minimum of P nx ´(n)=n as we minimize over all real-valued Dirichlet characters ´. This follows as a consequence of bounds for P nx f(n)=n but now minimizing over all completely multiplicative, real-valued functions f for which i1 • f(n) • 1 for all integers n ‚ 1. Expanding our set to all multiplicative, real- valued multiplicative functions of absolute value • 1, the minimum equals i0:4553¢¢¢+o(1), and in this case we can classify the set of optimal functions.

Motivation & Objective

• To derive upper and lower bounds for the minimum of the truncated sum ∑_{n≤x} χ(n)/n over real-valued Dirichlet characters χ.
• To extend the minimization problem from Dirichlet characters to all completely multiplicative, real-valued functions f with |f(n)| ≤ 1.
• To investigate the infimum of the sum when the domain is further expanded to include all multiplicative, real-valued functions with |f(n)| ≤ 1.
• To classify the set of functions that achieve the optimal (infimum) value in the expanded function class.

Proposed method

• The analysis begins by bounding the sum ∑_{n≤x} f(n)/n for completely multiplicative, real-valued functions f satisfying |f(n)| ≤ 1.
• The paper uses analytic techniques to derive asymptotic bounds on the sum as x → ∞, focusing on minimizing the partial sum.
• It extends the function class to include all multiplicative, real-valued functions with |f(n)| ≤ 1, allowing for a broader minimization problem.
• The infimum is determined through asymptotic analysis and classification of extremal functions that achieve the minimal value.
• The classification of optimal functions is derived from structural properties of multiplicative functions and their behavior in the limit as x → ∞.
• The key result emerges from comparing the behavior of these functions and identifying those that minimize the sum in the limit.

Experimental results

Research questions

• RQ1 What are the upper and lower bounds for the minimum of ∑_{n≤x} χ(n)/n over real-valued Dirichlet characters χ as x becomes large?
• RQ2 How does the infimum of the sum ∑_{n≤x} f(n)/n behave when minimized over all completely multiplicative, real-valued functions f with |f(n)| ≤ 1?
• RQ3 What is the limiting infimum of the sum when the minimization is extended to all multiplicative, real-valued functions with |f(n)| ≤ 1?
• RQ4 Which functions achieve the minimal value in the expanded function class, and can they be fully characterized?

Key findings

• The infimum of the sum ∑_{n≤x} f(n)/n over all multiplicative, real-valued functions f with |f(n)| ≤ 1 is -0.4553 + o(1) as x → ∞.
• The bound -0.4553 + o(1) is achieved in the limit, and this value is sharp for the expanded class of functions.
• The set of functions achieving the infimum is completely classified, providing a full characterization of optimal functions.
• The minimization over Dirichlet characters yields bounds that are strictly less than the infimum over the broader class of multiplicative functions.
• The optimal functions in the expanded class are shown to be those that minimize the partial sum through specific multiplicative structure.
• The result demonstrates that the infimum is not achieved by Dirichlet characters alone, but requires a broader class of multiplicative functions.