The Polya-Vinogradov Inequality bounds the partial sums of a nonprincipal Dirichlet character modulo a given number, showing that any run of consecutive terms of the character sums to a value no larger, in absolute terms, than a constant multiple of the square root of the modulus times its logarithm. Named for George Polya and Ivan Vinogradov, it is a standard tool of analytic number theory for controlling the size of character sums that appear throughout the study of primes in arithmetic progressions.
Facts
StatementFor any nonprincipal Dirichlet character chi(n) modulo q and any integers M and N, the sum of chi(n) for n from M+1 to M+N is O(sqrt(q) log q). 1 Connections
In Branch
Source Quadratic residue (Wikipedia)
Sources
1. Polya-Vinogradov inequality, Wikipedia
History sectionQuote, History section
The inequality was proved independently by Polya and Vinogradov in 1918.
Quadratic residue (Wikipedia)
In Branch: Number Theory, Lead sentenceQuote, In Branch: Number Theory, Lead sentence
In number theory, an integer q is a quadratic residue modulo n if it is congruent to a perfect square modulo n; that is, if there
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