Almost-all results on the p^\lambda problem

provided the condition (10) is satisfied.
We are now in a position to prove
Lemma 10: For every sufficiently sm<strong>all</strong> fixed η > 0 we have (11) when
N t+100η ≤ Y,
provided the conditions in (9) and (10) as well as the conditions
are satisfied.
t < (2 − λ)/5, t < (14 + 5λ)/54, t < (14 − 9λ)/32 (12)
Proof: We first note that the sum �
same manner as �
h≤H
h≤H
|Kh| 2 can alternatively be estimated in the
|Lh| 2 was estimated in Lemma 6. Thus, we also have (11) when
N t+100η ≤ Y ≤ N 2−λ−4t−100η ,
provided the conditions in (9) are satisfied. By this observation and Lemma 9, for proving
Lemma 10 it suffices to show that
and
(λ + 2t)/2 < min{1 − t, 7(1 + λ)/9 − 2t} (13)
max{(11λ + 36t)/7 − 2, t} < 2 − λ − 4t. (14)
Indeed, (13) follows from the first and the second condition in (12), and (14) follows
from the first and the third condition in (12). This completes the proof. ✷
5 Proof of Theorem 4
We suppose that 1/2 ≤ λ < 1 and
We choose t in such a manner that
and we take
0 ≤ τ < min{(2 − λ)/6, (14 − 9λ)/32}.
max{τ, 2(1 − λ)/5} < t < min{(2 − λ)/6, (14 − 9λ)/32},
u := N t+100η ,
v := N 2−λ−4t−100η ,
z := N t+100η .
8