Large character sums: pretentious characters and the Pólya-Vinogradov theorem.

*(English)*Zbl 1210.11090From the introduction: The best bound known for character sums was given independently by G. Pólya [Gött. Nachr. 1918, 21–29 (1918; JFM 46.0265.02)] and I. M. Vinogradov [J. Soc. Phys. Math. Univ. Perm. 2, 1–14 (1919; JFM 48.1352.04)]. For any non-principal Dirichlet character \(\chi\pmod q\) we let
\[
M(\chi):=\max_x\left|\sum_{n\leq x}\chi(n)\right|,
\]
and then the Pólya-Vinogradov inequality reads
\[
M(\chi) \ll \sqrt q \log q. \tag{1.1}
\]

There has been no subsequent improvement in this inequality other than in the implicit constant. Moreover it is believed that (1) will be difficult to improve since it is possible (though highly unlikely) that there is an infinite sequence of primes \(q\equiv 1\pmod 4\) for which \(({p\over q}) = 1\) for all \(p < q^{\varepsilon}\), in which case \(M(({\cdot\over q})) \gg_{\varepsilon} \sqrt q \log q\).

The unlikely possibility described above involves a quadratic character, and one might imagine that there are similar possibilities preventing one from improving (1.1) for higher order characters. Surprisingly, one of our main results shows that we can improve (1.1) for characters of odd, bounded order.

Theorem 1. If \(\chi\pmod q\) is a primitive character of odd order \(g\) then

\[ M(\chi) \ll_g \sqrt q(\log q)^{1-\frac{\delta_g}2 +o(1)}, \]

where \(\delta_g = (1- \frac g{\pi} \sin \frac{\pi}g)\).

The proof of Theorem 1 is based on some technical results which allow to characterize characters \(\chi\) for which \(M(\chi)\) is large. This characterization reveals that there is a hidden structure among the characters having large \(M(\chi)\).

In 1977 H. L. Montgomery and R. C. Vaughan [Invent. Math. 43, 69–82 (1977; Zbl 0362.10036)] showed if the Generalized Riemann Hypothesis is true then \[ M(\chi)\ll \sqrt q \log \log q.\tag{1.2} \]

This bound is best possible, up to the evaluation of the constant, in view of R. E. A. C. Paley’s 1932 result [J. Lond. Math. Soc. 7, 28–32 (1932; Zbl 0003.34101)] that there are infinitely many positive integers \(q\) such that \[ M\left(\left(\frac{\cdot}{q}\right)\right) \geq \left(\frac{e^\gamma}{\pi}+ o(1)\right)\sqrt q \log \log q,\tag{1.3} \] where \(\gamma = 0.5772\dots\) is the Euler-Mascheroni constant. Paley’s result gives large character sums for a thin class of carefully constructed quadratic characters, and one may ask if for each large prime \(q\) there are characters \(\chi \pmod q\) with similarly large \(M(\chi)\). The next result shows that there are indeed many such characters \(\chi\), and moreover one can point these character sums in any given direction.

Theorem 3. Let \(q\) be a large prime and let \(\theta\in (-\pi, \pi]\) be given. There is an absolute constant \(C_0\) such that for at least \(q^{1-C_0/(\log \log q)^2}\) characters \(\chi\pmod q\) with \(\chi(-1) = -1\) we have

\[ \sum_{n\leq x} \chi(n) = e^{i\theta} \frac{e^\gamma}{\pi}\sqrt q\left(\log\log q + O\left((\log\log q)^{1/2}\right)\right) \]

for all but \(o(q)\) natural numbers \(x\leq q\).

In view of Theorem 3 it may be surprising that there are analogues of Theorems 1 and 2 which give a sharper upper bound than (1.2) for characters of small odd order.

Theorem 4. Assume GRH. If \(\chi \pmod q\) is a primitive character of odd order \(g\) then \[ M(\chi) \ll_g \sqrt q(\log\log q)^{1-\frac{\delta_g}{2} +o(1)}. \]

On GRH, we can show that there exist arbitrarily large \(q\) and primitive characters \(\chi \pmod q\) of odd order \(g\) such that

\[ M(\chi) \gg_g \sqrt q(\log\log q)^{1\delta_g-o(1)}. \]

We believe that the exponent \(1-\delta_g = \frac g{\pi} \sin \frac{\pi}g\) in (1.4) is best possible, and that Theorem 4 can be improved to attain this bound.

Finally the authors use their previous bounds on \(L(1,\chi)\) [Q. J. Math. 53, No. 3, 265–284 (2002; Zbl 1022.11041)] to obtain an improvement of a theorem of A. J. Hildebrand [J. Number Theory 29, No. 3, 271–296 (1988; Zbl 0652.10029)] regarding the constant in the Pólya-Vinogradov theorem.

There has been no subsequent improvement in this inequality other than in the implicit constant. Moreover it is believed that (1) will be difficult to improve since it is possible (though highly unlikely) that there is an infinite sequence of primes \(q\equiv 1\pmod 4\) for which \(({p\over q}) = 1\) for all \(p < q^{\varepsilon}\), in which case \(M(({\cdot\over q})) \gg_{\varepsilon} \sqrt q \log q\).

The unlikely possibility described above involves a quadratic character, and one might imagine that there are similar possibilities preventing one from improving (1.1) for higher order characters. Surprisingly, one of our main results shows that we can improve (1.1) for characters of odd, bounded order.

Theorem 1. If \(\chi\pmod q\) is a primitive character of odd order \(g\) then

\[ M(\chi) \ll_g \sqrt q(\log q)^{1-\frac{\delta_g}2 +o(1)}, \]

where \(\delta_g = (1- \frac g{\pi} \sin \frac{\pi}g)\).

The proof of Theorem 1 is based on some technical results which allow to characterize characters \(\chi\) for which \(M(\chi)\) is large. This characterization reveals that there is a hidden structure among the characters having large \(M(\chi)\).

In 1977 H. L. Montgomery and R. C. Vaughan [Invent. Math. 43, 69–82 (1977; Zbl 0362.10036)] showed if the Generalized Riemann Hypothesis is true then \[ M(\chi)\ll \sqrt q \log \log q.\tag{1.2} \]

This bound is best possible, up to the evaluation of the constant, in view of R. E. A. C. Paley’s 1932 result [J. Lond. Math. Soc. 7, 28–32 (1932; Zbl 0003.34101)] that there are infinitely many positive integers \(q\) such that \[ M\left(\left(\frac{\cdot}{q}\right)\right) \geq \left(\frac{e^\gamma}{\pi}+ o(1)\right)\sqrt q \log \log q,\tag{1.3} \] where \(\gamma = 0.5772\dots\) is the Euler-Mascheroni constant. Paley’s result gives large character sums for a thin class of carefully constructed quadratic characters, and one may ask if for each large prime \(q\) there are characters \(\chi \pmod q\) with similarly large \(M(\chi)\). The next result shows that there are indeed many such characters \(\chi\), and moreover one can point these character sums in any given direction.

Theorem 3. Let \(q\) be a large prime and let \(\theta\in (-\pi, \pi]\) be given. There is an absolute constant \(C_0\) such that for at least \(q^{1-C_0/(\log \log q)^2}\) characters \(\chi\pmod q\) with \(\chi(-1) = -1\) we have

\[ \sum_{n\leq x} \chi(n) = e^{i\theta} \frac{e^\gamma}{\pi}\sqrt q\left(\log\log q + O\left((\log\log q)^{1/2}\right)\right) \]

for all but \(o(q)\) natural numbers \(x\leq q\).

In view of Theorem 3 it may be surprising that there are analogues of Theorems 1 and 2 which give a sharper upper bound than (1.2) for characters of small odd order.

Theorem 4. Assume GRH. If \(\chi \pmod q\) is a primitive character of odd order \(g\) then \[ M(\chi) \ll_g \sqrt q(\log\log q)^{1-\frac{\delta_g}{2} +o(1)}. \]

On GRH, we can show that there exist arbitrarily large \(q\) and primitive characters \(\chi \pmod q\) of odd order \(g\) such that

\[ M(\chi) \gg_g \sqrt q(\log\log q)^{1\delta_g-o(1)}. \]

We believe that the exponent \(1-\delta_g = \frac g{\pi} \sin \frac{\pi}g\) in (1.4) is best possible, and that Theorem 4 can be improved to attain this bound.

Finally the authors use their previous bounds on \(L(1,\chi)\) [Q. J. Math. 53, No. 3, 265–284 (2002; Zbl 1022.11041)] to obtain an improvement of a theorem of A. J. Hildebrand [J. Number Theory 29, No. 3, 271–296 (1988; Zbl 0652.10029)] regarding the constant in the Pólya-Vinogradov theorem.

Reviewer: Olaf Ninnemann (Berlin)

##### MSC:

11L40 | Estimates on character sums |

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\textit{A. Granville} and \textit{K. Soundararajan}, J. Am. Math. Soc. 20, No. 2, 357--384 (2007; Zbl 1210.11090)

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##### References:

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