MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

On the bottom of the page 399. of Iwaniec and Kowalski's Analytic Number Theory, the authors claim that $$h(t)=\int_{\mathbb H}k(i,z)y^s d\mu (z)$$ yields $$h(t)=\sqrt{\pi}\frac{\Gamma(s-\frac{1}{2})}{\Gamma(s+1)}R^s +O(R^{1/2}+S)$$ for $\frac{1}{2} < s\le 1$, where the implied constant depends on $s$. Can somebody show me at least the main steps of establishing the second equation from the first one?

share|cite|improve this question
Is the $\mathbb{R}$ correct or rather $\mathbb{H}$. – user9072 May 29 '13 at 22:14
it would help if you define what $k(i,z)$ is.I take it that $\mu$ is the measure on the upper half plane invariant under $SL_2({\mathbb R})$. – Venkataramana May 30 '13 at 2:11

Just a hint what is expected to be done here. The point pair invariant $k(w,z)$ comes from the function $k:\mathbb{R}^+\to\mathbb{C}$ such that $k(u)=1$ for $u\leq(R-2)/4$, $k(u)=0$ for $u\geq(R+S-2)/4$, and $k(u)$ is linear in between. The integral $h(t)$ is the Selberg transform of this function according to Lemma 15.6. Here we use that the function $y^s$ has Laplacian eigenvalue $s(1-s)$ with $s=1/2+it$, and at $y=1$ it equals $1$.

Now the task is to approximate $q(v)$, $g(r)$, $h(t)$, in this order, as given in Lemma 15.6.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.