Turán's method
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In mathematics, Turán's method provides lower bounds for exponential sums and complex power sums. The method has been applied to problems in equidistribution.
The method applies to sums of the form
where the b and z are complex numbers and ν runs over a range of integers. There are two main results, depending on the size of the complex numbers z.
Turán's first theorem[]
The first result applies to sums sν where for all n. For any range of ν of length N, say ν = M + 1, ..., M + N, there is some ν with |sν| at least c(M, N)|s0| where
The sum here may be replaced by the weaker but simpler .
We may deduce the Fabry gap theorem from this result.
Turán's second theorem[]
The second result applies to sums sν where for all n. Assume that the z are ordered in decreasing absolute value and scaled so that |z1| = 1. Then there is some ν with
See also[]
- Turán's theorem in graph theory
References[]
- Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Regional Conference Series in Mathematics. Vol. 84. Providence, RI: American Mathematical Society. ISBN 0-8218-0737-4. Zbl 0814.11001.
- Exponentials
- Analytic number theory