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By Andrew Granville
A method of analysis that plays a prominent role in Analytic Number Theory is the so-called circle method, which goes back to Hardy and Littlewood. This method uses the fact that, for any integer $n$,
\[\int_{0}^{1} \mathrm{e}^{2\mathrm{i}\pi nt}= \begin{cases}1 & \text{if}\ n=0, \\ 0 & \text{otherwise}.\end{cases}\]
For example, if we wish to count the number, $r(n)$, of solutions to the equation $p+q=n$ with $p$ and $q$ prime, we can express it as an integral as follows:
\[r(n)=\sum_{\substack{p,\,q \leqslant n \\ \text{both prime}}} \int_{0}^{1} \mathrm{e}^{2\mathrm{i}\pi(p+q-n)t}\,\mathrm{d}t=\int_0^1 \mathrm{e}^{-2\mathrm{i}\pi nt}\bigg(\sum_{p\,\text{prime},\,p\leqslant n}\mathrm{e}^{2\mathrm{i}\pi pt}\bigg)^2\mathrm{d}t.\]
The first equality holds because the integrand is $0$ when $p+q\neq n$ and $1$ otherwise, and the second is easy to check.
At first sight it looks more difficult to estimate the integral than it is to estimate $r(n)$ directly, but this is not the case. For instance, the prime number theorem for arithmetic progressions allows us to estimate $\sum_{p\leqslant n}\mathrm{e}^{2\mathrm{i}\pi pt}$ when $t$ is a rational $\ell/m$ with $m$ small. For in this case,
\[P\Big(\frac{\ell}{m}\Big)=\sum_{(a,m)=1}\mathrm{e}^{2\mathrm{i}\pi a \ell/m} \sum_{\substack{p\leqslant n,\\ p\equiv a \pmod m}}1 \approx \sum_{(a,m)=1}\mathrm{e}^{2\mathrm{i}\pi a \ell/m} \frac{\pi(n)}{\phi(n)}=\mu(m)\frac{\pi(n)}{\phi(n)}.\]
If $t$ is sufficiently close to $\ell/m$, then $P(t)\approx P(\ell/m)$; such values of $t$ are called the major arcs and we believe that the integral over the major arcs gives,in total,a very good approximation to $r(n)$.Thus to prove the Goldbach conjecture we need to show that the contribution to the integral from the other values of $t$ (that is, from the minor arcs) is small. In many problems one can successfully do this, but no one has yet succeeded in doing so for the Goldbach problem. Also useful is the “discrete analogue” of the above: using the identity
\[\frac{1}{m}\sum_{j=0}^{m-1}\mathrm{e}^{2\mathrm{i}\pi j n/m}=\begin{cases}1& \text{if $n\equiv 0\pmod m$},\\ 0& \text{ptherwise}.\end{cases}\]
(which holds for any given integer $m\geqslant 1$), we have that
\[r(n)=\sum_{\substack{p,\,q\leqslant n \\ \text{both prime}}}\frac{1}{m}\sum_{j=0}^{m-1}\mathrm{e}^{2\mathrm{i}\pi j(p+q-n)/m}=\sum_{j=0}^{m-1}\mathrm{e}^{-2\mathrm{i}\pi jn/m}P(j/m)^2\]
provided $m>n$. A similar analysis can be used here but working $\mod m$ sometimes has advantages, as it allows us to use properties of the multiplicative group $\mod m$.
Sums like $P(j/m)$ in the paragraph above or more simple sums like $\sum_{n\leqslant N}\mathrm{e}^{2\mathrm{i}\pi n^k/m}$ are called exponential sums. They play a central role in many of the calculations one does in analytic number theory. There are several techniques for investigating them.
(1) It is easy to sum the geometric progression $\sum_{n\leqslant N}\mathrm{e}^{2\mathrm{i}\pi n/m}$. with higher-degree polynomials one can often reduce to this case; for example, by writing $n_1-n_2=h$ we have
\[\bigg|\sum_{n\leqslant N}\mathrm{e}^{2\mathrm{i}\pi n^2/m}\bigg|^2=\sum_{n_1,\,n_2\leqslant N}\mathrm{e}^{2\mathrm{i}\pi(n_1^2-n_2^2)/m}=\sum_{|h|\leqslant N}\mathrm{e}^{2\mathrm{i}\pi h^2/m}\sum_{\max\{0,\,-h\}<n_2 \\ \leqslant \min\{N,\, N-h\}}\mathrm{e}^{4\mathrm{i}\pi hn_2/m},\]
and the inner sum is now a geometric progression.
(2) The work of Weil and Deligne, which gives very accurate results on the number of solutions to equations $\mod p$, is ideally suited to many applications in analytic number theory. For example, the “Kloosterman sum" $\sum_{a_1a_2\dotsm a_k\equiv b \pmod p}\mathrm{e}^{2\mathrm{i}\pi(a_1+a_2+\dotsb+a_k)/p}$, where the $a_i$ run over the integers $\mod p$ and $(b,q)=1$, appears naturally in many questions; Deligne showed that it has absolute value less than or equal to $kp(k−1)/2$, an extraordinary amount of cancellation in this sum which has about $p^{k−1}$ summands, each of absolute value 1.
(3) We discussed earlier the fact that the values of $\zeta(s)$ satisfy a symmetry about the line $\Re(s)=\frac{1}{2}$, given by the “functional equation.” There are other functions (called “modular functions”) that also have symmetries in the complex plane; typically the value of the function at $s$ is related to the value of the function at $(\alpha s+\beta)/(\gamma s+\delta)$, for some integers $\alpha,\beta,\gamma,\delta$ satisfying $\alpha\delta-\beta\gamma=1$. Sometimes an exponential sum can be related to the value of a modular function, and subsequently to the value of that modular function at another point, using the symmetry of the function.
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原文地址：http://www.cnblogs.com/pengdaoyi/p/4288076.html
