The Hasse–Davenport relations, introduced by , are two related identities for Gauss sums, one called the Hasse–Davenport lifting relation, and the other called the Hasse–Davenport product relation. The Hasse–Davenport lifting relation is an equality in number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

 relating Gauss sum

Gauss sum

In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typicallyG := G= \sum \chi\cdot \psi...

s over different fields. used it to calculate the zeta function of a Fermat hypersurface over a finite field, which motivated the Weil conjectures

Weil conjectures

In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....

.

Gauss sums are analogues of the gamma function

Gamma function

In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

 over finite fields, and the Hasse–Davenport product relation is the analogue of Gauss's multiplication formula
In fact the Hasse–Davenport product relation follows from the analogous multiplication formula for p-adic gamma functions together with the Gross–Koblitz formula

Gross–Koblitz formula

In mathematics the Gross–Koblitz formula, introduced by expresses a Gauss sum using a product of values of the p-adic gamma function. It is an analog of the Chowla–Selberg formula for the usual gamma function...

  of .

Hasse–Davenport lifting relation

Let F be a finite field with q elements, and F_s be the field such that [F_s:F] = s, that is, s is the dimension

Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...

 of the vector space F_s over F.

Let be an element of .

Let be a multiplicative

Multiplicative function

In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...

 character

Character (mathematics)

In mathematics, a character is a special kind of function from a group to a field . There are at least two distinct, but overlapping meanings...

 from F to the complex numbers.

Let be the norm from to defined by

Let
be the multiplicative character on which is the composition of with the norm

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

 from F_s to F, that is

Let ψ be some nontrivial additive character of F, and let
be the additive character on which is the composition of with the trace from F_s to F, that is
Let
be the Gauss sum

Gauss sum

In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typicallyG := G= \sum \chi\cdot \psi...

over F, and let
be the Gauss sum over .

Then the Hasse–Davenport lifting relation states that

Hasse–Davenport product relation

The Hasse–Davenport product relation states that
where ρ is a multiplicative character of exact order m dividing q–1 and χ is any multiplicative character and ψ is a non-trivial additive character.