A Variant of the Bombieri-vinogradov Theorem in Short Intervals with Applications
نویسنده
چکیده
We generalize the classical Bombieri-Vinogradov theorem to a short interval, 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 L/K of number fields. Using this result in conjunction with recent work of Maynard, we prove that rational primes in short intervals with a given splitting condition in a Galois extension L/Q exhibit dense clusters in short intervals. We explore several arithmetic applications related to questions of Serre regarding the nonvanishing Fourier coefficients of cuspidal modular forms, including finding dense clusters of fundamental discriminants d in short intervals for which the central values of d-quadratic twists of modular L-functions are non-vanishing.
منابع مشابه
A Bombieri-vinogradov Theorem for All Number Fields
Abstract. The classical theorem of Bombieri and Vinogradov is generalized to a non-abelian, non-Galois setting. This leads to a prime number theorem of “mixed-type” for arithmetic progressions “twisted” by splitting conditions in number fields. One can view this as an extension of earlier work of M. R. Murty and V. K. Murty on a variant of the Bombieri-Vinogradov theorem. We develop this theory...
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