Primes in Intervals and Semicircular Elements Induced by <i>p</i>-Adic Number Fields <inline-formula> <mml:math id="mm30000" display="block"> <mml:semantics> <mml:msub> <mml:mi mathvariant="double-struck">Q</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:semantics> </mml:math> </inline-formula> over Primes <i>p</i>

Abstract

In this paper, we study free probability on (weighted-)semicircular elements in a certain Banach *-probability space ( LS , τ 0 ) induced by measurable functions on p-adic number fields Q p over primes p . In particular, we are interested in the cases where such free-probabilistic information is affected by primes in given closed intervals of the set R of real numbers by defining suitable “truncated„ linear functionals on LS .

Keywords

• free probability
• primes
• p-adic number fields
• Banach *-probability spaces
• weighted-semicircular elements
• semicircular elements
• truncated linear functionals