Overview
This work details the development of a completely explicit Lerch, Chowla–Selberg formula for the Dedekind eta function, specifically structured without absolute values. The research also explores applications of this formula, particularly concerning the argument of individual Complex Multiplication (CM) values of the eta function. Additionally, the paper puts forth precise conjectures related to these CM values, supported by tabular data.
Research Context
The Dedekind eta function, denoted $\eta(\tau)$, is a fundamental object in the theory of modular forms. Its values at Complex Multiplication (CM) points, known as CM values, are of significant interest in number theory. The Lerch, Chowla–Selberg formula represents a known identity connecting special values of L-functions with values of the eta function. Previous formulations may have involved absolute values, necessitating a new explicit version without them.
Approach
The core approach involves deriving and presenting a completely explicit Lerch, Chowla–Selberg formula. The term 'explicit' implies that the formula is given in a form where all parameters and terms are precisely defined without ambiguity or reliance on implicit definitions. A key feature of this specific derivation is its formulation without the use of absolute values. Following the establishment of this formula, the researchers apply it to analyze the argument of individual CM values of the eta function. This application likely involves substituting CM points into the derived formula and examining the resulting properties of the argument. The work is also supported by the presentation of precise conjectures concerning these CM values, alongside a large table of supporting computational evidence.
Findings
- A completely explicit Lerch, Chowla–Selberg formula for the Dedekind eta function has been derived. This formula is notable for its presentation without absolute values.
- Applications of this formula extend to the argument of individual CM values of the eta function.
- Precise conjectures regarding these CM values have been developed.
- A large table has been compiled, providing supporting evidence for the presented conjectures.