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Results tagged with analytic-number-theory
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user 12705
On the blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.
19
votes
Accepted
Limit of $\frac{1}{n}\sum_{r=1}^n\frac{n\ (\mathrm{mod}\ r)}{r}$
This follows by elementary computation: we have
\begin{align}
\sum_{r=1}^n\frac{n\bmod r}r&=\sum_{r\le n}\frac nr-\sum_{r\le n}\left\lfloor\frac nr\right\rfloor\\\\
&=nH_n-|\{(r,s)\in\mathbb N^2:1\le …
- 47.1k
6
votes
Accepted
Clarification and Proof of Inequality (8.11) in Analytic Number Theory by Iwaniec and Kowalski
The assumptions imply
$$|S_f(N)|\le\sqrt2\frac N{\sqrt q}+2\sqrt q\log q.$$
Indeed, if $q\le2$, this follows from $|S_f(N)|\le N$; for $q\ge3$, we have $\log q>1$, thus
$$\begin{align*}
|S_f(N)|^2&\le …
- 47.1k