A Note on the Phragmen-Lindelof Theorem

This paper addresses errors in the literature surrounding the application of the Phragmén-Lindelöf principle, a crucial tool in analytic number theory for bounding L-functions by interpolating between vertical lines. Specifically, it focuses on correcting and generalizing Rademacher’s theorem, addressing inaccuracies in Trudgian’s generalization.

Significance: The Phragmén-Lindelöf principle is fundamental for bounding L-functions and has applications to many areas of analytic number theory. Correcting and improving existing versions are crucial for maintaining the rigor and accuracy of results derived from this principle. The work resolves issues with previous generalizations by Rademacher and Trudgian, aiming to ensure the validity of cited works dependent on these results.

Key Innovations:
1. Error Correction: The paper identifies and corrects an error in Trudgian’s lemma, ensuring the recoverability of proofs that use this result.

2. Generalized Version: The authors present a generalized version of Rademacher’s theorem, offering a simpler proof than the original.

3. Addressing Discrepancies: The work tackles inaccuracies arising from the handling of logarithmic terms, like log(Q+s), in the Phragmén-Lindelöf framework.

4. Iterated Logarithms: By introducing appropriate Q1 and Q2, the paper formulates a procedure for handling iterated logarithms.

Prior Ingredients:
* Phragmén-Lindelöf Principle: The core concept used to interpolate bounds of analytic functions within a strip, given bounds on the edges.
* Rademacher’s Theorem: An earlier general-purpose result that provides a foundation for the work.
* Analytic Number Theory: Basic understanding of L-functions and their bounding techniques.
* Complex Analysis: Fundamental knowledge of holomorphic functions, growth conditions, and maximum modulus principle.
* Trudgian’s Lemma: A prior generalization of Rademacher’s theorem that this paper aims to improve upon.