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.
Product identities in two variables x, q expand infinite products as infinite sums, which are linear combinations of theta functions; famous examples include Jacobi's triple product identity, Watson's quintuple identity, and Hirschhorn's septuple identity. We view these series expansions as representations in canonical bases of certain vector spaces of quasiperiodic meromorphic functions (related to sections of line and vector bundles), and find new identities for two nonuple products, an undecuple product, and several two-variable Rogers-Ramanujan type sums. Our main theorem explains a correspondence between the septuple product identity and the two original Rogers-Ramanujan identities, involving two-variable analogues of fifth-order mock theta functions. We also prove a similar correspondence between an octuple product identity of Ewell and two simpler variations of the Rogers-Ramanujan identities, which is related to third-order mock theta functions, and conjecture other occurrences of this phenomenon. As applications, we specialize our results to obtain identities for quotients of generalized Dedekind eta functions and mock theta functions.
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