Large character sums (Q2701704)

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 NEWLINENEWLINENEWLINEConjecture 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, NEWLINE\[NEWLINE\sum_{n\leq x} \chi(n) = \Psi(x,y;\chi)+o(\Psi(x,y;\chi_0)), \tag{*} NEWLINE\]NEWLINE where \(y=(\log q + \log^2 x)(\log\log q)^A\). NEWLINENEWLINENEWLINETheorem 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 \(\chi \pmod q\), NEWLINE\[NEWLINE\sum_{n\leq x} \chi(n) = \Psi(x,y;\chi) + O( \Psi(x,y;1)(\log\log q)^{-2}).NEWLINE\]NEWLINE Theorem 2 is a sharper version of Theorem 1 that is conditional on the General Riemann Hypothesis.NEWLINENEWLINENEWLINETheorem 3, which is too long to be fully quoted here, shows that in the range NEWLINE\[NEWLINEx\leq (\log \log q)^2 (\log \log \log q)^{-2},NEWLINE\]NEWLINE there are large character sums that point in any given direction. NEWLINENEWLINENEWLINETheorems 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 \(x=\log^A q\).NEWLINENEWLINENEWLINEThere 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 NEWLINE\[NEWLINE\max_{\chi\neq \chi_0} |\sum_{n\leq x} \chi(n)|\gg x\exp\left( -(1+o(1)) \log x \log \log x/\log \log q\right).NEWLINE\]NEWLINE 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.NEWLINENEWLINENEWLINETheorems 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 NEWLINE\[NEWLINE\sum_{n\leq x} (\frac{D}{n}) \gg x^{1/2} \exp((1+o(1)) \sqrt{\log q}/(\log \log q)).NEWLINE\]