We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet L-functions, we obtain an Ω±-result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ω-results for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes. 1.

Abstract. We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes. 1.

consists in replacing the computation [k]P by a multi-scalar multiplication with the form [k1]P + [k2]ψ(P), where the decomposition coe cients |k1|, |k2 | ≈ r1/2. Since the number of doublings is halved, this method potentially injects a significant speedup in the point multiplication computation on these elliptic curves. This approach might be generalized to m-dimensional case, which can achieve further speedups, if one could get higher degree decompositions with the form [k1]P + [k2]ψ(P) +... + [km]ψ(P) m−1 where |ki | ≈ r1/m. Constructing e ciently computable endomorphisms is one of the key problems in the GLV method. Gallant, Lambert and Vanstone gave some special examples in [10]. In 2002, Iijima, Matsuo, Chao and Tsujii [15] constructed an e cient computable homomorphism on elliptic curves E(Fp2) with j(E) ∈ Fp arising from the Frobenius map on a twist of E. Galbraith, Lin, and Scott [7,8] generalized their construction for a large class of elliptic curves over Fp2 (referred to as GLS curves) and applied the GLV method. They gave detailed implementations on these curves, showing that their method ran in between 0.70 and 0.84