53) (Colloquium Publications (Amer Mathematical Soc)) Hardcover – June 8, by Henryk Iwaniec (Author), Emmanuel Kowalski (Author). out of 5 stars 5. H. Iwaniec et E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI. In particular, when Iwaniec-Kowalski say that “slightly better results hold true for This means that the proof of Theorem in Iwaniec-Kowalski is incomplete.
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AMS :: Iwaniec and Kowalski: Analytic Number Theory
Here is a further improvement that proves Theorem 8. It is worthwhile to note that the improvement in “Update 2” is also needed to reach this conclusion. It is possible that with the improvements of 1 outlined above, one can reach 2 in a simpler way.
I have not examined this possibility for lack of time. The area above the blue curve and below the red curve is where the problem lies.
On page they mention the bound: You mean a book “Analytic Number Theory”?
By improving slightly the second and third terms in 1I was able to deduce 2 in all cases. See Updates 2 and 3 in my response. In short, Theorem 8. I am surprised that the authors left out a proof which needed careful and iwanuec details! The edit is fine.
Wait a few days and see if you get a more direct argument. If not, then you can open a new question, referring to this post and focusing on 2 only. The authors wanted to give a very explicit result, but this is of no importance, really.
I see you put a bounty on your question. As my response makes it clear that 1 and the trivial estimate alone are insufficient to imply kowalskkiyou should phrase the question more carefully. GH from MO I have a doubt. So that proves a weaker bound. You are right, I have been up for too long. Let me try to fix it or sleep on it.
Additional Material for the Book
I updated the argument. On the other hand, the number of exceptional pairs is finite as I explain in my post. The authors didn’t bother to prove the result or left it out for some other reason. So I was expecting a proof using the theory developed in the preceding section of Thm 8.
Well, in practice finitely many exceptions is not a problem. Note also that your 1 is already a consequence of a slightly sharper inequality. So there is iwanief room for improvement.
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