Mean Values of Character Sums

Library card.

---

H. L. Montgomery and R. C. Vaughan, "Mean Values of Character Sums," Canadian Journal of Mathematics 31 (1979), 476--487. Library card; DOI record.

Read status. The complete twelve-page Markdown copy was read. The theorem statements, proof mechanism, and the fourth-moment specialization below were checked against it; this is a source digest, not an independent verification of the proofs.

Mean-value results

For a nonprincipal Dirichlet character modulo $q$, set

$$M(\chi)=\max_N\left|\sum_{n=1}^N\chi(n)\right|.$$

Theorem 1 (printed p. 476) states that, for every fixed real $\kappa>0$,

$$\sum_{\chi\ne\chi_0}M(\chi)^{2\kappa} \ll_\kappa \phi(q)q^\kappa,$$

where the sum is over all nonprincipal characters modulo $q$. In particular, the fourth-moment case $\kappa=2$ is

$$\sum_{\chi\ne\chi_0}M(\chi)^4\ll \phi(q)q^2;$$

for prime $q$ this is $O(q^3)$. This fixed fourth moment is the result used in the proposed E0963 argument discussed below. The notation $\kappa$ here avoids confusion with the unrelated quantity $k=d(A)+1$ in that argument.

Theorem 2 (printed p. 476) gives the quadratic-character analogue averaged over prime moduli: for every fixed $\kappa>0$,

$$\sum_{2<p\leq P} \max_N\left|\sum_{n=1}^N\left(\frac np\right)\right|^{2\kappa} \ll_\kappa \pi(P)P^\kappa.$$

The corollary on the same page says that, for each $0<\theta<1$, a constant $C(\theta)$ makes $M(\chi)\leq C(\theta)q^{1/2}$ for at least $\theta\phi(q)$ nonprincipal characters modulo $q$, and makes the corresponding maximum Legendre-symbol sum at most $C(\theta)p^{1/2}$ for at least $\theta\pi(P)$ primes $p\leq P$. Neither Theorem 2 nor this typical-character corollary is the input used in the E0963 discussion; that use requires the all-character fourth-moment sum from Theorem 1.

Analytic mechanism

The proof first establishes the primitive-character estimate

$$\sum_\chi^*M(\chi)^{2\kappa}\ll\phi(q)q^\kappa \tag{11}$$

for integral $\kappa\geq2$, then sums over the primitive characters inducing characters modulo $q$ (printed p. 482, equation (11) and the display following it). Hölder monotonicity supplies all positive real moments from an unbounded sequence of integral ones.

The maximum over truncation points is handled by a Menchov--Rademacher dyadic decomposition (printed p. 482, equations (12)--(14)). Each dyadic block is converted by Pólya's Fourier expansion, Lemma 1 (printed p. 477), into a short Dirichlet polynomial with coefficients

$$a(h)\ll\min(2^{-r},h^{-1}) \tag{15}$$

after taking $H=q^{1/2}(\log q)^3$. Raising that polynomial to the $\kappa$th power gives equations (17)--(18) on printed p. 483,

$$\left(\sum_{0<h\leq H}\chi(h)e(h\nu2^{-r})a(h)\right)^\kappa =\sum_{n\leq H^\kappa}\chi(n)b(n), \qquad b(n)\ll d_\kappa(n)\min(2^{-\kappa r},n^{-1}).$$

Character orthogonality, Lemma 3 (printed p. 477), bounds the resulting second moment of this Dirichlet polynomial. Summation over the dyadic scale $r$ and the possible dyadic endpoint $\nu$ then proves equation (16), hence (11). The proof of Theorem 2 (printed pp. 483--486) replaces full character orthogonality by the quadratic-character estimates of Lemmas 6 and 9, uses Burgess's short-interval estimate from Lemma 2 to discretize the maximizing endpoint, and splits the Fourier frequencies according to equation (22).

Fourth-moment translation used toward E0963

A discussion of Problem 963, whose capture is not held here, proposes the following application. Let $q$ be prime and let $A,B\subseteq (\mathbb Z/q\mathbb Z)^*$, with $B$ an arithmetic progression. For uniformly random $r\in(\mathbb Z/q\mathbb Z)^*$, character orthogonality gives

$$\operatorname{Var}|rA\cap B| =\frac{1}{(q-1)^2}\sum_{\chi\ne\chi_0} \left|\sum_{a\in A}\chi(a)\right|^2 \left|\sum_{b\in B}\chi(b)\right|^2.$$

After multiplying by the inverse of its nonzero common difference, a modular progression is a cyclic interval, so

$$\left|\sum_{b\in B}\chi(b)\right|\leq2M(\chi).$$

Theorem 1 with $\kappa=2$ therefore gives $\sum_{\chi\ne\chi_0}|\sum_{b\in B}\chi(b)|^4\ll q^3$. The trivial pointwise bound followed by Parseval gives

$$\sum_{\chi\ne\chi_0}\left|\sum_{a\in A}\chi(a)\right|^4 \leq |A|^2\sum_\chi\left|\sum_{a\in A}\chi(a)\right|^2 =(q-1)|A|^3.$$

Cauchy--Schwarz consequently yields

$$\operatorname{Var}|rA\cap B|\ll |A|^{3/2}.$$

Since $\mathbb E|rA\cap B|=|A||B|/(q-1)$, Chebyshev gives the lower-tail estimate

$$\Pr\!\left( |rA\cap B|\leq\frac{|A||B|}{2(q-1)} \right) \ll\frac{(q-1)^2}{|A|^{1/2}|B|^2}.$$

Thus, under the discussion's parameter hypotheses $|A|\geq L^{10}$ and $|B|\geq q/L$, the failure probability is $O(L^{-3})$. This is the precise bridge from Montgomery--Vaughan's moment theorem to the proposed equidistribution lemma for multiplicative dilates.

Limitations for distinct subset sums

The paper contains no theorem about dissociated sets or distinct subset sums. Its fourth-moment bound proves only the analytic estimate in the preceding section. In particular, it does not establish the discussion's asserted real-to-integer reduction, the passage from populated progression cells to a dissociated lift, the endpoint correction needed to prevent wraparound modulo $q$, the recursion for the positive-integer extremal function, or the iteration claimed to yield $(1-o(1))\log_2 n$. Those are separate steps in an informal forum proof whose corrected full argument is not present in this source.

The bound also does not prove the exact conjecture $f(n)\geq\lfloor\log_2n\rfloor$, and it gives no direct information about the largest dissociated subset of an arbitrary set of reals. Its implied constant may depend on the moment parameter; the proposed use fixes $\kappa=2$, so no uniformity in growing moments is available or needed for that particular second-moment calculation.