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| In [[mathematics]], a '''character sum''' is a sum
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| :<math>\Sigma \chi(n)\,</math>
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| of values of a [[Dirichlet character]] χ ''[[Modular arithmetic|modulo]]'' ''N'', taken over a given range of values of ''n''. Such sums are basic in a number of questions, for example in the distribution of [[quadratic residues]], and in particular in the classical question of finding an upper bound for the [[least quadratic non-residue]] ''modulo'' ''N''. Character sums are often closely linked to [[exponential sum]]s by the [[Gauss sum]]s (this is like a finite [[Mellin transform]]).
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| Assume χ is a nonprincipal Dirichlet character to the modulus ''N''.
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| ==Sums over ranges==
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| The sum taken over all residue classes mod ''N'' is then zero. This means that the cases of interest will be sums <math>\Sigma</math> over relatively short ranges, of length ''R'' < ''N'' say, | |
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| :<math>M \le n < M + R.</math>
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| A fundamental improvement on the trivial estimate <math>\Sigma = O(N)</math> is the '''[[quadratic residues#The P.C3.B3lya.E2.80.93Vinogradov inequality|Pólya–Vinogradov inequality]]''' ([[George Pólya]], [[I. M. Vinogradov]], independently in 1918), stating in [[big O notation]] that
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| :<math>\Sigma = O(\sqrt{N}\log N).</math>
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| Assuming the [[generalized Riemann hypothesis]], [[Hugh Montgomery (mathematician)|Hugh Montgomery]] and [[Robert Charles Vaughan (mathematician)|R. C. Vaughan]] have shown<ref>Montgomery and Vaughan (1977)</ref> that there is the further improvement
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| :<math>\Sigma = O(\sqrt{N}\log\log N).</math>
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| ==Summing polynomials==
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| Another significant type of character sum is that formed by
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| :<math>\Sigma \chi(F(n))\,</math>
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| for some function ''F'', generally a [[polynomial]]. A classical result is the case of a quadratic, for example,
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| :<math>F(n) = n(n + 1)\,</math> | |
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| and χ a [[Legendre symbol]]. Here the sum can be evaluated (as −1), a result that is connected to the [[local zeta-function]] of a [[conic section]].
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| More generally, such sums for the [[Jacobi symbol]] relate to local zeta-functions of [[elliptic curve]]s and [[hyperelliptic curve]]s; this means that by means of [[André Weil]]'s results, for ''N'' = ''p'' a [[prime|prime number]], there are non-trivial bounds
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| :<math>O(\sqrt{p}).</math> | |
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| The constant implicit in the notation is [[linear function|linear]] in the [[genus (mathematics)|genus]] of the curve in question, and so (Legendre symbol or hyperelliptic case) can be taken as the degree of ''F''. (More general results, for other values of ''N'', can be obtained starting from there.)
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| Weil's results also led to the ''Burgess bound'',<ref>Burgess (1957)</ref> applying to give non-trivial results beyond Pólya–Vinogradov, for ''R'' a power of ''N'' greater than 1/4.
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| Assume the modulus ''N'' is a prime.
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| :<math>
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| \begin{align}
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| \Sigma & \ll p^{1/2} \log p , \\[6pt]
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| \Sigma & \ll 2 R^{1/2} p^{3/16} \log p , \\[6pt]
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| \Sigma & \ll r R^{1-1/r} p^{(r+1)/4r^2} (\log p)^{1/2r}
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| \end{align}
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| </math>
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| for any integer ''r'' ≥ 3.<ref>Montgomery and Vaughan (2007), p.315</ref>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{cite journal | author=G. Pólya | authorlink=George Pólya | title=Ueber die Verteilung der quadratischen Reste und Nichtreste | journal=Nachr. Akad. Wiss. Goettingen | year=1918 | pages=21–29 | jfm=46.0265.02 }}
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| * {{cite journal | author=I. M. Vinogradov | authorlink=Ivan Matveyevich Vinogradov | title=Sur la distribution des residus and nonresidus des puissances | journal=J. Soc. Phys. Math. Univ. Permi | year=1918 | pages=18–28 | jfm=48.1352.04 }}
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| * {{cite journal | doi=10.1112/S0025579300001157 | author=D. A. Burgess | title=The distribution of quadratic residues and non-residues | journal=Mathematika | volume=4 | issue=02 | year=1957 | pages=106–112 | zbl=0081.27101 }}
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| * {{cite journal | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) | coauthors=[[Robert Charles Vaughan (mathematician)|Robert C. Vaughan]] | title=Exponential sums with multiplicative coefficients | journal=Invent. Math. | volume=43 | year=1977 | issue=1 | pages=69–82 | doi=10.1007/BF01390204 | zbl=0362.10036 }}
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| * {{cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) | coauthors=Robert C. Vaughan | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | publisher=[[Cambridge University Press]] | year=2007 | isbn=0-521-84903-9 | zbl=1142.11001 | pages=306–325}}
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| ==Further reading==
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| * {{cite book | zbl=0754.11022 | last=Korobov | first=N.M. | title=Exponential sums and their applications | others=Translated from the Russian by Yu. N. Shakhov | series=Mathematics and Its Applications (Soviet Series) | volume=80 | location=Dordrecht | publisher=Kluwer Academic Publishers | year=1992 | isbn=0-7923-1647-9 }}
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| ==External links==
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| *{{MathWorld|urlname=Polya-VinogradovInequality| title=The Pólya–Vinogradov inequality}}
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| *[http://planetmath.org/encyclopedia/PolyaVinogradovInequality.html PlanetMath article on the Pólya–Vinogradov inequality]
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| [[Category:Analytic number theory]]
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