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This monograph explores the relationships and connections among number theory, operator theory, operator algebra theory, and representation theory within free-probabilistic frameworks. Arithmetic functions are examined as Krein-space operators under suitable representation, extending to free probability through the determination of point-evaluation linear functionals for primes.

Graduate students and researchers with interests in the aforementioned mathematical fields or related applied scientific areas will discover novel approaches to understanding these topics.

Understanding the images of arithmetic functions at primes as forms of matrices or operators acting Krein spaces will be of special interest to readers looking for new methods to analyse them mathematically.