Exponential sum

An exponential sum is a finite sum of complex exponentials whose phases depend on arithmetic data. Such sums encode arithmetic structure through cancellation among terms.

Additive character: A homomorphism from an additive group to the complex numbers of absolute value one. Additive characters provide the oscillatory terms in many arithmetic sums.
Gauss sum: A finite sum of a multiplicative character evaluated at residues, weighted by an additive character. Quadratic phases yield a classical, explicitly evaluable family.
Weyl's criterion: A criterion stating that a sequence is uniformly distributed modulo one exactly when its nonzero Fourier averages tend to zero. Bounds on exponential sums test equidistribution of arithmetic sequences.
Geometric series: A finite sum whose successive terms have a constant ratio. Linear phases reduce to geometric series with explicit formulas.
Multiplicative character: A homomorphism from a multiplicative group to the complex numbers of absolute value one. These characters supply arithmetic phases in sums over residue classes.
Kloosterman sum: A finite sum of additive characters applied to a variable and its multiplicative inverse modulo an integer. Its inverse-variable phase creates a central example of difficult cancellation.
Prime number theorem: A theorem describing the asymptotic density of prime numbers among positive integers. Exponential-sum estimates support analytic approaches to prime distribution.
Dirichlet kernel: The trigonometric polynomial formed by summing consecutive Fourier modes from negative to positive indices. Its structured linear phases behave differently from irregular arithmetic phases.
Root of unity: A complex number whose positive integer power equals one. Finite-group characters often take root-of-unity values.
Weyl sum: A sum of complex phases formed from a polynomial evaluated at consecutive integers. Polynomial phases show how differencing can control cancellation.

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