TY - JOUR

T1 - A variant of the Bombieri-Vinogradov theorem in short intervals and some questions of Serre

AU - Thorner, Jesse

N1 - Funding Information:
This work was supported by the National Natural Science Foundation under Grant No. 50575144, 60474036, and supported by Key Foundation Program of Shanghai Sciences & Technology Committee under Grant No. . The author wish to acknowledge the relative study works finished by Dr. Chen Wenjie, Dr. Zhou Lv, Dr. Chen Xizhang, Dr. Zhu Zhenyou and Mr. Lang Yuyou.
Publisher Copyright:
© Cambridge Philosophical Society 2016.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - We generalise the classical Bombieri-Vinogradov theorem for short intervals to a non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are twisted by a splitting condition in a Galois extension of number fields. Using this result in conjunction with the recent work of Maynard, we prove that rational primes with a given splitting condition in a Galois extension L/ exhibit bounded gaps in short intervals. We explore several arithmetic applications related to questions of Serre regarding the non-vanishing Fourier coefficients of cuspidal modular forms. One such application is that for a given modular L-function L(s, f), the fundamental discriminants d for which the d-quadratic twist of L(s, f) has a non-vanishing central critical value exhibit bounded gaps in short intervals.

AB - We generalise the classical Bombieri-Vinogradov theorem for short intervals to a non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are twisted by a splitting condition in a Galois extension of number fields. Using this result in conjunction with the recent work of Maynard, we prove that rational primes with a given splitting condition in a Galois extension L/ exhibit bounded gaps in short intervals. We explore several arithmetic applications related to questions of Serre regarding the non-vanishing Fourier coefficients of cuspidal modular forms. One such application is that for a given modular L-function L(s, f), the fundamental discriminants d for which the d-quadratic twist of L(s, f) has a non-vanishing central critical value exhibit bounded gaps in short intervals.

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U2 - 10.1017/S0305004116000050

DO - 10.1017/S0305004116000050

M3 - Article

AN - SCOPUS:84959114754

SN - 0305-0041

VL - 161

SP - 53

EP - 63

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

IS - 1

ER -