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Large character sums. (English) Zbl 0983.11053
A central problem of analytic number theory is the behavior of the sums $$\sum_{n\leq x}\chi(n)$$, where $$\chi$$ is a Dirichlet character. This important paper is a substantial addition to our understanding of these character sums. The authors make a detailed study of character sums over “smooth” integers. Let $$f$$ be a multiplicative function, and define $$\Psi(x,y;f)=\sum_{n\leq x, p|n \rightarrow p\leq y} f(n)$$. Their study is motivated by the belief that character sums can be large only because of extraordinary behavior of $$\chi(p)$$ for small primes $$p$$. This belief is formalized into
Conjecture 1. There exists a constant $$A>0$$ such that for any non-principal character $$\chi \pmod q$$ and for any $$1\leq x \leq q$$ we have, uniformly, $\sum_{n\leq x} \chi(n) = \Psi(x,y;\chi)+o(\Psi(x,y;\chi_0)), \tag{*}$ where $$y=(\log q + \log^2 x)(\log\log q)^A$$.
Theorem 1 is an unconditional “almost all” approximation to Conjecture 1. Assume that $$1\leq x\leq q$$ and $$y\geq \log q \log x (\log \log q)^5$$. Then for all but at most $$q^{1-1/\log x}$$ characters $$\chi \pmod q$$, $\sum_{n\leq x} \chi(n) = \Psi(x,y;\chi) + O( \Psi(x,y;1)(\log\log q)^{-2}).$ Theorem 2 is a sharper version of Theorem 1 that is conditional on the General Riemann Hypothesis.
Theorem 3, which is too long to be fully quoted here, shows that in the range $x\leq (\log \log q)^2 (\log \log \log q)^{-2},$ there are large character sums that point in any given direction.
Theorems 2 and 3 combine to give the conditional Corollary A. Assume the Riemann Hypothesis for $$L(s,\chi)$$. Then the estimate (*) holds if $$\log x/\log\log q\to \infty$$ as $$q\to\infty$$. This is “best possible” in the sense that, for any given $$A>0$$, for every prime $$q$$ there exists a non-principal character $$\chi \pmod q$$ such that $$|\sum_{n\leq x} \chi(n)|\gg_A x$$, where $$x=\log^A q$$.
There are several other theorems that establish the existence of large values of character sums. Theorems 4-7 and Corollaries 1-3 apply to general non-principal characters. To indicate the flavor of these results, we quote Corollary 1. If $$\log x \geq (\log\log q)^2$$, then $\max_{\chi\neq \chi_0} |\sum_{n\leq x} \chi(n)|\gg x\exp\left( -(1+o(1)) \log x \log \log x/\log \log q\right).$ If, in addition, $$q$$ has at most $$(\log q)^{1-\epsilon}$$ distinct prime factors, then this bound holds in the extended range $$\log x/\log\log q\to\infty$$. The proofs of Theorems 4-7 and Corollaries 1-3 ultimately depend on lower bounds for certain divisor sums, and as the authors note, these results may be generalized to a much wider class of arithmetic functions.
Theorems 9-11 deal with real characters. A nice example is Theorem 10, which states the following. Suppose that $$q$$ is large and $$\exp((\log q)^{1/2}) \leq x \leq q/\exp((\log q)^{1/2}).$$ Then there exist fundamental discriminants $$D$$ in the range $$q\leq D \leq 2q$$ with $\sum_{n\leq x} (\frac{D}{n}) \gg x^{1/2} \exp((1+o(1)) \sqrt{\log q}/(\log \log q)).$

##### MSC:
 11L40 Estimates on character sums 11N25 Distribution of integers with specified multiplicative constraints
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##### References:
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