Character sums ============== .. if-builder:: html .. toctree:: :maxdepth: 2 1. Explicit Polya-Vinogradov inequalities ----------------------------------------- The main Theorem of :cite:`Qiu91` implies the following result. .. admonition:: Theorem (1991) :class: thm-tme-emt For :math:`\chi` a primitive character to the modulus :math:`q > 1`, we have :math:`\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 :math:`\chi` is not especially primitive, but is still non-principal, we have :math:`\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 :cite:`Bachman-Rachakonda01` into the following. .. admonition:: Theorem (2001) :class: thm-tme-emt For :math:`\chi` a non-principal character to the modulus :math:`q > 1`, we have :math:`\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 These results are superseded by :cite:`Frolenkov11` and more recently by :cite:`Frolenkov-Soundararajan13` into the following. .. admonition:: Theorem (2013) :class: thm-tme-emt For :math:`\chi` a non-principal character to the modulus :math:`q\ge 1000`, we have :math:`\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 :cite:`Pomerance11` and get the following. .. admonition:: Theorem (2013) :class: thm-tme-emt For :math:`\chi` a primitive character to the modulus :math:`q \ge 1200`, we have :math:`\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 :math:`q\ge40`. In case :math:`\chi` odd, the constant :math:`1/(2\pi)` has already been asymptotically obtained in :cite:`Landau18-3`. When :math:`\chi` is odd and :math:`M=1`, the best asymptotical constant before 2020 was :math:`1/(3\pi)` from Theorem 7 of :cite:`Granville-Soundararajan07`, In case :math:`\chi` even, we have :math:`\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 :math:`23/(35\pi\sqrt{3})` from Theorem 7 of :cite:`Granville-Soundararajan07`. These results are improved upon for large values squarefree values of :math:`q` in :cite:`Bordignon-Kerr20` by a different method into the following. .. admonition:: Theorem (2020) :class: thm-tme-emt For :math:`\chi` a primitive character to the squarefree modulus :math:`q \ge \exp(1088\ell^2)`, we have :math:`\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 :math:`q\ge40`. Corresponding estimates when :math:`q` is not squarefree are proved in :cite:`Bordignon21`, the saving :math:`1/4` being slightly degraded to :math:`3/8`. 2. Burgess type estimates -------------------------- The following from :cite:`Trevino15-2` is an explicit version of Burgess with the only restriction being :math:`p\ge 10^7`. .. admonition:: Theorem (2015) :class: thm-tme-emt Let :math:`p` be a prime such that :math:`p \ge 10^7`. Let :math:`\chi` be a non-principal character :math:`\bmod{\,p}`. Let :math:`r` be a positive integer, and let :math:`M` and :math:`N` be non-negative integers with :math:`N\ge 1`. Then :math:`\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. .. admonition:: Theorem (2015) :class: thm-tme-emt Let :math:`p` be a prime. Let :math:`\chi` be a non-principal character :math:`\bmod{\,p}`. Let :math:`M` and :math:`N` be non-negative integers with :math:`N\ge 1`, let :math:`2\le r\le 10` be a positive integer, and let :math:`p_0` be a positive real number. Then for :math:`p \ge p_0`, there exists :math:`c_1(r)`, a constant depending on :math:`r` and :math:`p_0` such that :math:`\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 :math:`c_1(r)` is given by :math:`r` :math:`p_0=10^7` :math:`p_0=10^{10}` :math:`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 :math:`\log` if we restrict the range of :math:`N` or if we have :math:`r\ge 3`. .. admonition:: Theorem (2015) :class: thm-tme-emt Let :math:`p` be a prime. Let :math:`\chi` be a non-principal character :math:`\bmod{\,p}`. Let :math:`M` and :math:`N` be non-negative integers with :math:`1\le N\le2 p^{\frac{1}{2} + \frac{1}{4r}}` or :math:`r\ge 3`. Let :math:`r\le 10` be a positive integer, and let :math:`p_0` be a positive real number. Then for :math:`p \ge p_0`, there exists :math:`c_2(r)`, a constant depending on :math:`r` and :math:`p_0` such that :math:`\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 :math:`c_2(r)` is given by :math:`r` :math:`p_0=10^7` :math:`p_0=10^{10}` :math:`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 :cite:`McGown12` has slightly worse constants in a slightly larger range of :math:`N` for smaller values of :math:`p`. .. admonition:: Theorem (2012) :class: thm-tme-emt Let :math:`p\ge 2\cdot 10^{4}` be a prime number. Let :math:`M` and :math:`N` be non-negative integers with :math:`1\le N\le 4 p^{\frac{1}{2} +\frac{1}{4r}}`. Suppose :math:`\chi` is a non-principal character :math:`\bmod{\,p}`. Then there exists a computable constant :math:`C(r)` such that :math:`\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 :math:`C(r)` is given by :math:`r` :math:`C(r)` :math:`r` :math:`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 :cite:`Booker06`. .. admonition:: Theorem (2006) :class: thm-tme-emt Let :math:`p > 10^{20}` be a prime number with :math:`p \equiv 1 \pmod{4}`. Let :math:`r\in \{2,3,4,\ldots,15\}`. Let :math:`M` and :math:`N` be real numbers such that :math:`0 < M , N \le 2\sqrt{p}`. Let :math:`\chi` be a non-principal quadratic character :math:`\bmod{\,p}`. Then :math:`\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 :math:`\alpha(r)` and :math:`\beta(r)` are given by :math:`r` :math:`\alpha(r)` :math:`\beta(r)` :math:`r` :math:`\alpha(r)` :math:`\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 :cite:`Jain-Sharma-Khale-Liu21` . .. admonition:: Theorem (2021) :class: thm-tme-emt Let :math:`\chi` be a primitive character with modulus :math:`q\ge e^{e^{9.594}}`. Then for :math:`N\le q^{5/8}`, we have :math:`\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)}}.`