Imaginary Powers of the Dedekind Eta Function

ABSTRACT

In this article, complex powers of the Dedekind eta function are studied. The vanishing of the nth Fourier coefficients are labeled by the roots of an attached polynomial p_n(x). We study these polynomials, and their values and roots distribution. The considered polynomials of degree n ⩽ 700 are verified as Hurwitz polynomials. We study the value distribution of the polynomials restricted to the imaginary axis.

KEYWORDS:

• Fourier coefficients
• Euler products
• integer-valued polynomials
• Lehmer conjecture
• roots of polynomials

2010 AMS SUBJECT CLASSIFICATION:

• Primary 26C10, 11F20
• Secondary 11F30, 11F37, 12D10