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I've been studying Davenport's Multiplicative number theory and recently came across the concept of major arcs and minor arcs while proving Vinogradov's Theorem. While I understand their utility in estimating exponential sums and dealing with the problem, I'm curious about their motivations behind their introduction. How they thought about this idea to deal with the quantity $S(\alpha)$. We have

$$S(\alpha)=\sum_{\substack{k \leq N}}\Lambda(k) e(k \alpha)$$ and it is represented as an integral over $[0,1]$ using Fourier coefficent formula given by $$r(N) = \int_{0}^{1}S(\alpha)^3 e(-N\alpha) d\alpha $$ Could someone provide insights into how mathematicians initially got the idea of major arcs and minor arcs? What other problems or challenges they aimed to address by using this concept? I'd also be interested in learning about the key figures and milestones in the development of this framework.

Any references or books on this topic would be greatly appreciated. Thank you!

zero2infinity
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Originally, there were no major arcs or minor arcs in circle method. Instead, Hardy and Ramanujan used Farey dissection. Let

$$ {a'\over q'}<\frac aq<{a''\over q''} $$

be consecutive Farey fractions of order $N$. Then it can be proven that if

$$ I_{a/q}=\left[{\frac aq}-{1\over q(q+q')},\frac aq+{1\over q(q+q'')}\right)\tag2 $$

and $I_{0/1}=[0,1/(N+1))\cup[1-1/(N+1),1)$, then $I_{a/q}$ partitions $[0,1)$ completely:

$$ [0,1)=\bigcup_{\substack{0\le a<q\le N\\(a,q)=1}}I_{a/q}. $$

This setup allows Hardy and Ramanujan to successfully deduce an asymptotic series for the partition function $p(n)$.

Hardy-Littlewood's majoar and minor arcs

The concept of major/minor arcs was introduced in Hardy and Littlewood's investigation of Waring's method. In their investigation, they realized that the asymptotic for integrals over $I_{a/q}$ can be conveniently estimated when $q$ is small (say $q\le N_1$).

However, when $N_1<q\le N$, instead of doing asymptotic expansion, they used Weyl's inequality to derive upper bounds, so these integrals essentially go inside the error terms. As a result, the Farey arcs $I_{a/q}$ satisfying $N_1<q\le N$ are considered minor arcs, with those satisfying $q\le N_1$ called major arcs.

Vinogradov's improvement

Subsequently, it was realized that the delicate structure of $I_{a/q}$ in (2) in unnecessary in the major/minor-arc version of the circle method.

In the works of Hardy, Littlewood, and Ramanujan, circle methods were applied to infinite series. As a result, in order to guarantee convergence, the integration had to happen in a circle of radius less than unity. However, Vinogradov found that these series can be replaced with truncated exponential sums, so we can integrate on the unit circle without issues:

$$ \sum_{m_1+m_2+\dots+m_s=n}a_{m_1}a_{m_2}\cdots a_{m_s}=\int_0^1F^s(\alpha)e^{-2\pi in\alpha}\mathrm d\alpha,\quad F(\alpha)=\sum_{m\le x}a_me^{2\pi im\alpha},\quad n\le x\quad s\in\mathbb N $$

In $I_{a/q}$ the goal is that the integrand $F^s(\alpha)e^{-2\pi in\alpha}$ can be well approximated by itself with $\alpha=a/q$, so Vinogradov decided that the only requirement for major arcs is to ensure $\alpha$ is sufficiently close to $a/q$:

$$ I'_{a/q}=\left\{\alpha:\left|\alpha-\frac aq\right|\le\frac1Q\right\}, $$

and we choose $N$ and $Q$ appropriately so that different $I'_{a/q}$'s are always disjoint. Consequently, the minor arcs, in Vinogradov's setup, is defined by

$$ [0,1)\setminus\bigcup_{\substack{0\le a<q\le N\\(a,q)=1}}I'_{a/q}. $$

This is all the historical development of major arcs and minor arcs in circle methods.

Although irrelevant to this question, it should be noted that Kloosterman had also made an important contribution to Hardy-Littlewood circle method (that can also be thought as a variant of major/minor arc approach). I have written a detailed account here.

TravorLZH
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