×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] P.T. Bateman and S. Chowla, Averages of character sums, Proc. Amer. Math. Soc 1 (1950), 781-787. · Zbl 0041.01904
[2] D. A. Burgess, The distribution of quadratic residues and non-residues, Mathematika 4 (1957), 106 – 112. · Zbl 0081.27101 · doi:10.1112/S0025579300001157 · doi.org
[3] Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. · Zbl 0453.10002
[4] J. B. Friedlander and H. Iwaniec, A note on character sums, The Rademacher legacy to mathematics (University Park, PA, 1992) Contemp. Math., vol. 166, Amer. Math. Soc., Providence, RI, 1994, pp. 295 – 299. · Zbl 0820.11049 · doi:10.1090/conm/166/01632 · doi.org
[5] S. W. Graham and C. J. Ringrose, Lower bounds for least quadratic nonresidues, Analytic number theory (Allerton Park, IL, 1989) Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 269 – 309.
[6] A. Granville and K. Soundararajan, The spectrum of multiplicative functions (to appear). · Zbl 1036.11042
[7] A. Granville and K. Soundararajan, The distribution of \(L(1,\chi )\) (to appear). · Zbl 1188.11039
[8] G.H. Hardy and S. Ramanujan, The normal number of prime factors of a number \(n\), Quart. J. Math 48 (1917), 76-92. · JFM 46.0262.03
[9] Adolf Hildebrand, A note on Burgess’ character sum estimate, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), no. 1, 35 – 37. · Zbl 0589.10039
[10] Adolf Hildebrand and Gérald Tenenbaum, Integers without large prime factors, J. Théor. Nombres Bordeaux 5 (1993), no. 2, 411 – 484. · Zbl 0797.11070
[11] Hugh L. Montgomery, An exponential polynomial formed with the Legendre symbol, Acta Arith. 37 (1980), 375 – 380. · Zbl 0369.10024
[12] Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. · Zbl 0814.11001
[13] H. L. Montgomery and R. C. Vaughan, Exponential sums with multiplicative coefficients, Invent. Math. 43 (1977), no. 1, 69 – 82. · Zbl 0362.10036 · doi:10.1007/BF01390204 · doi.org
[14] R.E.A.C. Paley, A theorem on characters, J. London Math. Soc 7 (1932), 28-32. · Zbl 0003.34101
[15] Carl Pomerance, On the distribution of round numbers, Number theory (Ootacamund, 1984) Lecture Notes in Math., vol. 1122, Springer, Berlin, 1985, pp. 173 – 200. · Zbl 0565.10038 · doi:10.1007/BFb0075761 · doi.org
[16] G. Tenenbaum, Cribler les entiers sans grand facteur premier, Philos. Trans. Roy. Soc. London Ser. A 345 (1993), no. 1676, 377 – 384 (French, with English and French summaries). · Zbl 0795.11042 · doi:10.1098/rsta.1993.0136 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.