Piatetski-shapir prime number theorem and chebotarev density theorem
Güloğlu, Ahmet Muhtar
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Let K be a nite Galois extension of the eld Q of rational numbers. In this thesis, we derive an asymptotic formula for the number of the Piatetski-Shapiro primes not exceeding a given quantity for which the associated Frobenius class of automorphisms coincide with any given conjugacy class in the Galois group of K=Q. Applying this theorem to appropriate eld extensions, we conclude that there are in nitely many Piatetski-Shapiro primes lying in a given arithmetic progresion and furthermore there are in nitely many primes that can be expressed as a sum of a square and a xed positive integer multiple of another square.
KeywordsChebotarev density theorem
Piatetski-Shapiro prime number theorem
Exponential sums over ideals