We generalize results of Alladi, Dawsey, and Sweeting and Woo for Chebotarev densities to general densities of sets of primes. We show that if K is a number field and S is any set of prime ideals with natural density δ(S) within the primes, thenwhere µ(a) is the generalized Möbius function and D(K, S) is the set of integral ideals a ⊆ OK with unique prime divisor of minimal norm lying in S. Our result can be applied to give formulas for densities of various sets of prime numbers, including those lying in a Sato-Tate interval of a fixed elliptic curve, and those in Beatty sequences such as ⌊πn⌋.
Fix a non-CM elliptic curve E/Q, and let aE(p) = p + 1 − #E(Fp) denote the trace of Frobenius at p. The Sato-Tate conjecture gives the limiting distribution µST of aE(p)/(2 √ p)within [−1, 1]. We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval I ⊆ [−1, 1], let pI,n denote the nth prime such that aE(p)/(2 √ p) ∈ I.We show lim infn→∞(pI,n+m −pI,n) < ∞ for all m ≥ 1 for "most" intervals, and in particular, for all I with µST (I) ≥ 0.36. Furthermore, we prove a common generalization of our bounded gap result with the Green-Tao theorem. To obtain these results, we demonstrate a Bombieri-Vinogradov type theorem for Sato-Tate primes.
Let S(a, b) denote the normalized Dedekind sum. We study the range of possible values for S(a, b) = k q with gcd(k, q) = 1. Girstmair proved local restrictions on k depending on q (mod 12) and whether q is a square and conjectured that these are the only restrictions possible. We verify the conjecture in the cases q even, q a square divisible by 3 or 5, and 2 ≤ q ≤ 200 (the latter by computer), and provide progress towards a general approach.
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