Almost-all results on the p^\lambda problem

More documents

Recommendations

Info

Theorem 2: Let B > 0 and λ ∈ (0, 1/2) be given. Define ⎧ ⎪⎨ 5/12 − λ/6 if λ ≤ 1/8, H(λ) := � � ⎪⎩ 1 + λ/2 �2 /3 − λ if 1/8 < λ ≤ 1/2. Then the asymptotic formula (1) holds true for almost <strong>all</strong> real θ with respect to the Lebesgue measure if τ is any fixed real number lying in the range 0 ≤ τ < H(λ). In the present paper we use a completely different approach to prove the following almost-<strong>all</strong> result for the range 1/2 ≤ λ < 1. Theorem 3: Let B > 0 and λ ∈ [1/2, 1) be given. Define H(λ) := min{(2 − λ)/6, (14 − 9λ)/32}. Then the asymptotic formula (1) holds true for almost <strong>all</strong> real θ with respect to the Lebesgue measure if τ is any fixed real number lying in the range 0 ≤ τ < H(λ). We note that the function H(λ) defined in Theorems 2 and 3 is continuous on the interval (0, 1) and that for λ0 = 1/2 we have H(λ0) = 1/4 = (1 − λ0)/2. For λ0 = 1/2 Theorem 3 therefore does’nt provide any improvement beyond the result of Theorem 1. However, for any λ ∈ (1/2, 1) the τ-range in Theorem 3 is wider than that in Theorem 1. In [3] we improved the result of Theorem 3 for single θ’s in the range 59/85 = 0.694... < λ < 1. There we proved that in this range the exponent −(1 − λ)/2 + ε in Theorem 1 may be replaced by −G(λ) + ε, where G(λ) := min{max{(35 − 22λ)/129, 1/7}, 5/18 − λ/6}. Even this result is exceeded by that of Theorem 3 for almost-<strong>all</strong> θ since in the range 59/85 < λ < 1 we have H(λ) > G(λ). In a similar manner like [2, Theorem 2] was derived from [2, Proposition], we sh<strong>all</strong> later derive the assertion of Theorem 3 from Theorem 4: Let the conditions of Theorem 3 be kept, and suppose that C > 0. Then we have as N → ∞. �1 0 � � � � � � � � � � � n ≤ N, � n λ − θ � < n −τ Λ(n) − N 1−τ 1 − τ � �2 � � � � � � � � dθ ≪ N 2(1−τ ) (log N) C Acknowledgements. This research has been supported by a Marie Curie Fellowship of the European Community programme “Improving the Human Research Potential and the Socio-Economic Knowledge Base” under contract number HPMF-CT-2002- 02157. 2 (2)
2 Reduction to exponential sums We sh<strong>all</strong> employ a similar method like in [3], where we linked the p λ problem with Piatetski-Shapiro primes, that is, primes of the form [n c ] with fixed c > 1 and n ∈ �. A standard procedure for attacking problems related to Piatetski-Shapiro primes is as follows (see [11] for instance): Firstly, one reduces the sum of Λ(n) in question (in our paper, this is the sum on the left-hand side of (1)) to exponential sums of the form � n∼N Λ(n)e � hn λ� . Secondly, one breaks up these exponential sums into certain bilinear exponential sums (so-c<strong>all</strong>ed type I and type II sums) by using Vaughan-type identities. Fin<strong>all</strong>y, one estimates these bilinear exponential sums in a non-trivial way. Here we also follow this approach. However, since the first two steps were completely carried out in several papers (see [11] for example), we omit some details in these steps. Throughout the following, we suppose that λ, τ and t are given real numbers satisfying 1/2 ≤ λ < 1 and 0 ≤ τ < t < 1, and we put H := N t . The parameter t will later be choosen suitably subject to λ and τ. Furthermore, we suppose that η is a (sm<strong>all</strong>) positive real number. By the notation k ∼ K we mean k to lie in some interval K1 ≤ k ≤ K2 with K/2 ≤ K1 ≤ K2 ≤ 2K. Using the Fourier series expansion of {x}, we obtain Lemma 1: Suppose that (ah) is a sequence of complex numbers. Assume that we uniformly have, as N → ∞, � ah � 1≤h≤H N
Page 1: Almost-all
Page 5 and 6: We shall prove the
Page 7 and 8: � 2 Kh ≪ (log N) N 1−λ h −
Page 9 and 10: Then the conditions in (9), (10), (
Page 11: Using this inequality, (22) and Tay

Almost-all results on the p^\lambda problem

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?