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Let K be a finite Galois extension of the field ℚ of rational numbers. We prove an asymptotic formula for the number of Piatetski-Shapiro primes not exceeding a given quantity for which the associated Frobenius class of automorphisms coincides with any given conjugacy class in the Galois group of K/ℚ. In particular, this shows that there are infinitely many Piatetski-Shapiro primes of the form a² + nb² for any given natural number n.
Yıldırım Akbal, and Ahmet Muhtar Güloğlu. "Piatetski-Shapiro meets Chebotarev." Acta Arithmetica 167.4 (2015): 301-325. <http://eudml.org/doc/278969>.
@article{YıldırımAkbal2015, abstract = {Let K be a finite Galois extension of the field ℚ of rational numbers. We prove an asymptotic formula for the number of Piatetski-Shapiro primes not exceeding a given quantity for which the associated Frobenius class of automorphisms coincides with any given conjugacy class in the Galois group of K/ℚ. In particular, this shows that there are infinitely many Piatetski-Shapiro primes of the form a² + nb² for any given natural number n.}, author = {Yıldırım Akbal, Ahmet Muhtar Güloğlu}, journal = {Acta Arithmetica}, keywords = {Chebotarev's density theorem; Piatetski-Shapiro prime number theorem; exponential sums over ideals; generalized Vaughan's identity; van der Corput's method; Vinogradov's method}, language = {eng}, number = {4}, pages = {301-325}, title = {Piatetski-Shapiro meets Chebotarev}, url = {http://eudml.org/doc/278969}, volume = {167}, year = {2015}, }
TY - JOUR AU - Yıldırım Akbal AU - Ahmet Muhtar Güloğlu TI - Piatetski-Shapiro meets Chebotarev JO - Acta Arithmetica PY - 2015 VL - 167 IS - 4 SP - 301 EP - 325 AB - Let K be a finite Galois extension of the field ℚ of rational numbers. We prove an asymptotic formula for the number of Piatetski-Shapiro primes not exceeding a given quantity for which the associated Frobenius class of automorphisms coincides with any given conjugacy class in the Galois group of K/ℚ. In particular, this shows that there are infinitely many Piatetski-Shapiro primes of the form a² + nb² for any given natural number n. LA - eng KW - Chebotarev's density theorem; Piatetski-Shapiro prime number theorem; exponential sums over ideals; generalized Vaughan's identity; van der Corput's method; Vinogradov's method UR - http://eudml.org/doc/278969 ER -