Exponential Sums and Character Sums

Exponential and character sums are the primary technical tools for estimating the distribution of arithmetic sequences. Bounds on these sums translate directly into results about primes, divisor functions, and other arithmetic objects.

Exponential Sums

Definition6.3Exponential Sum Estimates

A complete exponential sum (or Ramanujan sum) is $c_q(n) = \sum_{\substack{a=1 \\ (a,q)=1}}^q e^{2\pi i an/q}$ , which equals $\mu(q/(q,n))\phi(q)/\phi(q/(q,n))$ . Incomplete exponential sums $S = \sum_{M < n \leq M+N} e^{2\pi i f(n)}$ are bounded by the Weyl bound, the van der Corput method, or Vinogradov's method depending on the nature of $f$ .

Definition6.4Weyl Sums

For a polynomial $f(x) = \alpha_k x^k + \cdots + \alpha_1 x + \alpha_0$ of degree $k$ , the Weyl sum is $S = \sum_{n=1}^N e^{2\pi i f(n)}$ . Weyl's inequality (via differencing): $|S|^{2^{k-1}} \ll N^{2^{k-1}} \left(\frac{1}{N} + \frac{1}{q} + \frac{q}{N^k}\right)$ where $|\alpha_k - a/q| \leq 1/(qN^k)$ . The bound is non-trivial when $\alpha_k$ is not too well approximated by rationals with small denominator.

Character Sum Bounds

ExamplePolya-Vinogradov Inequality

For a non-principal character $\chi$ mod $q$ : $\left|\sum_{n=M+1}^{M+N} \chi(n)\right| \leq c\sqrt{q}\log q$ . This bound is independent of $N$ and $M$ . The constant $c$ can be taken as $2/\pi^2$ for primitive characters. The Polya-Vinogradov inequality is essentially best possible: Paley showed that for infinitely many $q$ , $\max_N |\sum_{n=1}^N \chi(n)| \gg \sqrt{q}\log\log q$ for the Legendre symbol.

RemarkBurgess Bound

Burgess's theorem: For a primitive character $\chi$ mod $q$ and any $\varepsilon > 0$ : $\left|\sum_{n=M+1}^{M+N} \chi(n)\right| \ll N^{1-1/r} q^{(r+1)/(4r^2)+\varepsilon}$ for any integer $r \geq 1$ . This is non-trivial for $N > q^{1/4+\varepsilon}$ , beating the Polya-Vinogradov range of $N > q^{1/2+\varepsilon}$ . Burgess's bound is the sharpest known for character sums over short intervals and implies the least quadratic non-residue mod $p$ is $O(p^{1/(4\sqrt{e})+\varepsilon})$ .

Kloosterman Sums

Definition6.5Kloosterman Sums and Weil's Bound

The Kloosterman sum is $S(m, n; c) = \sum_{\substack{x \bmod c \\ (x,c)=1}} e^{2\pi i(mx + n\bar{x})/c}$ where $\bar{x}$ is the multiplicative inverse of $x$ mod $c$ . Weil's bound: $|S(m,n;c)| \leq d(c) \sqrt{c} \cdot \gcd(m,n,c)^{1/2}$ . Kloosterman sums arise naturally in the Fourier expansion of Poincare series and in the Kuznetsov trace formula, connecting spectral theory of automorphic forms to exponential sums.