A note on the zeroes of the Fredholm series

TL;DR

This paper proves that the Fredholm series has infinitely many zeros in the unit disk, and every complex number appears as a value infinitely often, resolving a longstanding open question.

Contribution

It establishes that the Fredholm series attains every complex value infinitely often within the unit disk, confirming the conjecture about its zeros.

Findings

Fredholm series has infinitely many zeros in the unit disk

Every complex number is attained infinitely many times as a value of the series

Addresses a longstanding open question in the theory of Fredholm series

Abstract

The issue had been raised whether the Fredholm series $z+z^2+...+z^{2^n}+...$ has infinitely many zeroes in the unit disk. We provide an affirmative answer, proving that in fact every complex number occurs as a value infinitely many times.