Character sums¶
1. Explicit Polya-Vinogradov inequalities¶
The main Theorem of [Qiu, 1991] implies the following result.
Theorem (1991)
For \(\chi\) a primitive character to the modulus \(q > 1\), we have \(\left|\sum\limits_{a=M+1}^{M+N}\chi(a)\right|\le\frac{4}{\pi^2}\sqrt{q}\log q+0.38\sqrt{q}+\frac{0.637}{\sqrt{q}}\).
When \(\chi\) is not especially primitive, but is still non-principal, we have \(\left|\sum\limits_{a=M+1}^{M+N}\chi(a)\right|\le\frac{8\sqrt{6}}{3\pi^2}\sqrt{q}\log q+0.63\sqrt{q}+\frac{1.05}{\sqrt{q}}\).
This was improved later by [Bachman and Rachakonda, 2001] into the following.
Theorem (2001)
For \(\chi\) a non-principal character to the modulus \(q > 1\), we have \(\left|\sum\limits_{a=M+1}^{M+N}\chi(a)\right|\le\frac{1}{3\log 3}\sqrt{q}\log q+6.5\sqrt{q}\).
These results are superseded by [Frolenkov, 2011] and more recently by [Frolenkov and Soundararajan, 2013] into the following.
Theorem (2013)
For \(\chi\) a non-principal character to the modulus \(q\ge 1000\), we have \(\left|\sum\limits_{a=M+1}^{M+N}\chi(a)\right|\le\frac{1}{\pi\sqrt{2}}\sqrt{q}(\log q+6)+\sqrt{q}\).
In the same paper they improve upon estimates of [Pomerance, 2011] and get the following.
Theorem (2013)
For \(\chi\) a primitive character to the modulus \(q \ge 1200\), we have \(\displaystyle \max_{M,N}\left|\sum_{a=M+1}^{M+N}\chi(a)\right|\le\begin{cases}\frac{2}{\pi^2}\sqrt{q}\log q+\sqrt{q},& \chi\text{ even,}\\\frac{1}{2\pi}\sqrt{q}\log q+\sqrt{q},& \chi\text{ odd}.\end{cases}\) This latter estimates holds as soon as \(q\ge40\).
In case \(\chi\) odd, the constant \(1/(2\pi)\) has already been asymptotically obtained in [Landau, 1918]. When \(\chi\) is odd and \(M=1\), the best asymptotical constant before 2020 was \(1/(3\pi)\) from Theorem 7 of [Granville and Soundararajan, 2007], In case \(\chi\) even, we have
\(\displaystyle \max_{M,N}\left|\sum_{a=M}^N\chi(a)\right|=2\max_{N}\left|\sum_{a=1}^N\chi(a)\right|.\) (The LHS is always less than the RHS. Equality is then easily proved). The asymptotical best constant in 2007 was \(23/(35\pi\sqrt{3})\) from Theorem 7 of [Granville and Soundararajan, 2007].
These results are improved upon for large values squarefree values of \(q\) in [Bordignon and Kerr, 2020] by a different method into the following.
Theorem (2020)
For \(\chi\) a primitive character to the squarefree modulus \(q \ge \exp(1088\ell^2)\), we have \(\displaystyle \max_{N}\left|\sum_{a=1}^{N}\chi(a)\right|\le\begin{cases} \frac{2}{\pi^2}\sqrt{q}\bigl(\frac14+\frac{1}{4\ell}\bigr)\log q +\bigl(49+\frac{1}{1088\ell}\bigr)\sqrt{q},& \chi\text{ even,}\\ \frac{1}{2\pi}\bigl(\frac12+\frac{1}{2\ell}\bigr)\sqrt{q}\log q +\bigl(49+\frac{1}{1088\ell}\bigr)\sqrt{q},& \chi\text{ odd}.\end{cases}\) This latter estimates holds as soon as \(q\ge40\).
Corresponding estimates when \(q\) is not squarefree are proved in [Bordignon, 2021], the saving \(1/4\) being slightly degraded to \(3/8\).
2. Burgess type estimates¶
The following from [Treviño, 2015] is an explicit version of Burgess with the only restriction being \(p\ge 10^7\).
Theorem (2015)
Let \(p\) be a prime such that \(p \ge 10^7\). Let \(\chi\) be a non-principal character \(\bmod{\,p}\). Let \(r\) be a positive integer, and let \(M\) and \(N\) be non-negative integers with \(N\ge 1\). Then \(\displaystyle \left|\sum_{a=M+1}^{M+N}\chi(a)\right|\le 2.74 N^{1-\frac{1}{r}}p^{\frac{r+1}{4r^2}}(\log{p})^{\frac{1}{r}}.\)
From the same paper, we get the following more specific result.
Theorem (2015)
Let \(p\) be a prime. Let \(\chi\) be a non-principal character \(\bmod{\,p}\). Let \(M\) and \(N\) be non-negative integers with \(N\ge 1\), let \(2\le r\le 10\) be a positive integer, and let \(p_0\) be a positive real number. Then for \(p \ge p_0\), there exists \(c_1(r)\), a constant depending on \(r\) and \(p_0\) such that \(\displaystyle \left|\sum_{a=M+1}^{M+N}\chi(a)\right|\le c_1(r) N^{1-\frac{1}{r}} p^{\frac{r+1}{4r^2}}(\log{p})^{\frac{1}{r}}\) where \(c_1(r)\) is given by
\(r\) \(p_0=10^7\) \(p_0=10^{10}\) \(p_0=10^{20}\)
2 2.7381 2.5173 2.3549
3 2.0197 1.7385 1.3695
4 1.7308 1.5151 1.3104
5 1.6107 1.4572 1.2987
6 1.5482 1.4274 1.2901
7 1.5052 1.4042 1.2813
8 1.4703 1.3846 1.2729
9 1.4411 1.3662 1.2641
10 1.4160 1.3495 1.2562
We can get a smaller exponent on \(\log\) if we restrict the range of \(N\) or if we have \(r\ge 3\).
Theorem (2015)
Let \(p\) be a prime. Let \(\chi\) be a non-principal character \(\bmod{\,p}\). Let \(M\) and \(N\) be non-negative integers with \(1\le N\le2 p^{\frac{1}{2} + \frac{1}{4r}}\) or \(r\ge 3\). Let \(r\le 10\) be a positive integer, and let \(p_0\) be a positive real number. Then for \(p \ge p_0\), there exists \(c_2(r)\), a constant depending on \(r\) and \(p_0\) such that \(\displaystyle \left|\sum_{a=M+1}^{M+N}\chi(a)\right|\le c_2(r) N^{1-\frac{1}{r}} p^{\frac{r+1}{4r^2}}(\log{p})^{\frac{1}{2r}},\) where \(c_2(r)\) is given by
\(r\) \(p_0=10^7\) \(p_0=10^{10}\) \(p_0=10^{20}\)
2 3.7451 3.5700 3.5341
3 2.7436 2.5814 2.4936
4 2.3200 2.1901 2.1071
5 2.0881 1.9831 1.9037
6 1.9373 1.8504 1.7748
7 1.8293 1.7559 1.6843
8 1.7461 1.6836 1.6167
9 1.6802 1.6262 1.5638
10 1.6260 1.5786 1.5210
Kevin McGown in [McGown, 2012] has slightly worse constants in a slightly larger range of \(N\) for smaller values of \(p\).
Theorem (2012)
Let \(p\ge 2\cdot 10^{4}\) be a prime number. Let \(M\) and \(N\) be non-negative integers with \(1\le N\le 4 p^{\frac{1}{2} +\frac{1}{4r}}\). Suppose \(\chi\) is a non-principal character \(\bmod{\,p}\). Then there exists a computable constant \(C(r)\) such that \(\displaystyle \left|\sum_{a=M+1}^{M+N}\chi(a)\right|\le C(r) N^{1-\frac{1}{r}} p^{\frac{r+1}{4r^2}}(\log{p})^{\frac{1}{2r}},\) where \(C(r)\) is given by
\(r\) \(C(r)\) \(r\) \(C(r)\)
2 10.0366 9 2.1467
3 4.9539 10 2.0492
4 3.6493 11 1.9712
5 3.0356 12 1.9073
6 2.6765 13 1.8540
7 2.4400 14 1.8088
8 2.2721 15 1.7700
If the character is quadratic (and with a more restrictive range), we have slightly stronger results due to Booker in [Booker, 2006].
Theorem (2006)
Let \(p > 10^{20}\) be a prime number with \(p \equiv 1 \pmod{4}\). Let \(r\in \{2,3,4,\ldots,15\}\). Let \(M\) and \(N\) be real numbers such that \(0 < M , N \le 2\sqrt{p}\). Let \(\chi\) be a non-principal quadratic character \(\bmod{\,p}\). Then \(\displaystyle \left|\sum_{a=M+1}^{M+N}\chi(a)\right|\le \alpha(r) N^{1-\frac{1}{r}} p^{\frac{r+1}{4r^2}}\left(\log{p} + \beta(r)\right)^{\frac{1}{2r}},\) where \(\alpha(r)\) and \(\beta(r)\) are given by
\(r\) \(\alpha(r)\) \(\beta(r)\) \(r\) \(\alpha(r)\) \(\beta(r)\)
2 1.8221 8.9077 9 1.4548 0.0085
3 1.8000 5.3948 10 1.4231 -0.4106
4 1.7263 3.6658 11 1.3958 -0.7848
5 1.6526 2.5405 12 1.3721 -1.1232
6 1.5892 1.7059 13 1.3512 -1.4323
7 1.5363 1.0405 14 1.3328 -1.7169
8 1.4921 0.4856 15 1.3164 -1.9808
Concerning composite moduli, we have the next result in [Jain-Sharma et al., 2021] .
Theorem (2021)
Let \(\chi\) be a primitive character with modulus \(q\ge e^{e^{9.594}}\). Then for \(N\le q^{5/8}\), we have \(\displaystyle \left|\sum_{a=M+1}^{M+N}\chi(a)\right| \le 9.07 \sqrt{N}q^{3/16}(\log q)^{1/4} \bigl(2^{\omega(q)}d(q)\bigr)^{3/4} \sqrt{\frac{q}{\varphi(q)}}.\)