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Statement

Lemma (printed p. 380, unnumbered). Let $q\in\mathbb N$, let $(A_n)$ be a sequence of complex numbers with period $q$, and write

$$\hat A_m=\sum_{n=1}^{q}A_n\,e\Bigl(\frac{mn}{q}\Bigr), \qquad e(t)=\exp(2\pi it).$$

Then for all integers $a<b$,

$$\Bigl|\sum_{a<n\le b}A_n-\frac{b-a}{q}\hat A_0\Bigr| \le(\log q)\max_{1\le m<q}|\hat A_m|.$$

There is no restriction on the length $b-a$ in the lemma itself.

Inequality (1) (printed p. 381). For an interval $I$ of length at most $q$ and an integer $c$, write $S_I(q;c)=\sum_{n\in I}e(c\bar n/q)$, where $n\bar n\equiv1\pmod q$ and the terms with $(n,q)>1$ are omitted, and let

$$S(m,c;q)=\sum_{n=1}^{q}e\Bigl(\frac{mn+c\bar n}{q}\Bigr)$$

be the Kloosterman sum, with the same omission. The article states that the lemma, applied to $S_I(q;c)$ under this length assumption, gives

$$|S_I(q;c)|\le(1+\log q)\max_{1\le m\le q}|S(m,c;q)|.$$

Source. D. R. Heath-Brown, Arithmetic applications of Kloosterman sums, Nieuw Arch. Wiskd. (5) 1 (2000), no. 4, 380–384; the lemma on printed p. 380, inequality (1) and the definition of $S(m,c;q)$ on printed p. 381. The edition is identified on the source card.

Read depth. Claims checked: the lemma and inequality (1) were read clause by clause against the print. The article gives no proof of either; nothing here is independently reviewed.

Proof pointer

The article states the lemma without proof, as the standard device for converting an incomplete sum into complete ones. The usual argument expands $A_n$ by Fourier inversion, $A_n=q^{-1}\sum_{m=0}^{q-1}\hat A_m e(-mn/q)$; the frequency $m=0$ gives the main term, and each other frequency contributes a geometric series over $a<n\le b$, bounded by $1/(2\|m/q\|)$, whose sum over $1\le m<q$, divided by $q$, is at most $\log q$. For (1), the sequence equal to $e(c\bar n/q)$ on residues prime to $q$ and $0$ elsewhere has $\hat A_m=S(m,c;q)$; the main term $\frac{b-a}{q}S(0,c;q)$ is at most one complete sum in size when the interval has length at most $q$, which accounts for the $1$ in $1+\log q$.

Dependencies

None in the article.

Bears on

• Problem 445: the lemma, with (1) and Weil's bound, is the input to the origin-rectangle estimate on p. 382. The lemma holds for every interval $a<n\le b$, but the article applies it only to the origin rectangle and states no translated-interval result.