In mathematics, Turán's method provides lower bounds for

exponential sum In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function :e(x) = \exp(2\pi ix).\, Therefore, a typic ...

s and complex power sums. The method has been applied to problems in equidistribution. The method applies to sums of the form :

s_\nu = \sum_^N b_n z_n^\nu \

where the ''b'' and ''z'' are

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s 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 ''s''_ν where

, z_n,  \ge 1

for all ''n''. For any range of ''ν'' of length ''N'', say ''ν'' = ''M'' + 1, ..., ''M'' + ''N'', there is some ''ν'' with , ''s''_''ν'', at least ''c''(''M'', ''N''), ''s''₀, where :

c(M,N) = \left(\right)^ \ .

The sum here may be replaced by the weaker but simpler

\left(\right)^

. We may deduce the

Fabry gap theorem In mathematics, the Fabry gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a certain "gap" between them. Such a power series is "badly behaved" in the sense that it cann ...

from this result.

Turán's second theorem

The second result applies to sums ''s''_ν where

, z_n,  \le 1

for all ''n''. Assume that the ''z'' are ordered in decreasing absolute value and scaled so that , ''z''₁, = 1. Then there is some ν with :

, s_\nu,  \ge 2 \left(\right)^N \min_ \left\vert\right\vert \ .

References

* {{DEFAULTSORT:Turan's method Exponentials Analytic number theory

Turán's first theorem

Turán's second theorem

See also

References