Distribution (number theory)

The multiplication theorem for the Hurwitz zeta function

${\displaystyle \zeta (s,a)=\sum _{n=0}^{\infty }(n+a)^{-s}}$ 

gives a distribution relation

${\displaystyle \sum _{p=0}^{q-1}\zeta (s,a+p/q)=q^{s}\,\zeta (s,qa)\ .}$ 

Hence for given s, the map ${\displaystyle t\mapsto \zeta (s,\{t\})}$  is a distribution on Q/Z.

Recall that the Bernoulli polynomials B_n are defined by

${\displaystyle B_{n}(x)=\sum _{k=0}^{n}{n \choose n-k}b_{k}x^{n-k}\ ,}$ 

for n ≥ 0, where b_k are the Bernoulli numbers, with generating function

${\displaystyle {\frac {te^{xt}}{e^{t}-1}}=\sum _{n=0}^{\infty }B_{n}(x){\frac {t^{n}}{n!}}\ .}$ 

They satisfy the distribution relation

${\displaystyle B_{k}(x)=n^{k-1}\sum _{a=0}^{n-1}b_{k}\left({\frac {x+a}{n}}\right)\ .}$ 

Thus the map

${\displaystyle \phi _{n}:{\frac {1}{n}}\mathbb {Z} /\mathbb {Z} \rightarrow \mathbb {Q} }$ 

defined by

${\displaystyle \phi _{n}:x\mapsto n^{k-1}B_{k}(\langle x\rangle )}$ 

is a distribution.^[4]

The cyclotomic units satisfy distribution relations. Let a be an element of Q/Z prime to p and let g_a denote exp(2πia)−1. Then for a≠ 0 we have^[5]

${\displaystyle \prod _{pb=a}g_{b}=g_{a}\ .}$ 

One considers the distributions on Z with values in some abelian group V and seek the "universal" or most general distribution possible.

Let h be an ordinary distribution on Q/Z taking values in a field F. Let G(N) denote the multiplicative group of Z/N‌Z, and for any function f on G(N) we extend f to a function on Z/N‌Z by taking f to be zero off G(N). Define an element of the group algebra F[G(N)] by

${\displaystyle g_{N}(r)={\frac {1}{|G(N)|}}\sum _{a\in G(N)}h\left({\left\langle {\frac {ra}{N}}\right\rangle }\right)\sigma _{a}^{-1}\ .}$ 

The group algebras form a projective system with limit X. Then the functions g_N form a distribution on Q/Z with values in X, the Stickelberger distribution associated with h.

Consider the special case when the value group V of a distribution φ on X takes values in a local field K, finite over Q_p, or more generally, in a finite-dimensional p-adic Banach space W over K, with valuation |·|. We call φ a measure if |φ| is bounded on compact open subsets of X.^[6] Let D be the ring of integers of K and L a lattice in W, that is, a free D-submodule of W with K⊗L = W. Up to scaling a measure may be taken to have values in L.

Let D be a fixed integer prime to p and consider Z_D, the limit of the system Z/pⁿD. Consider any eigenfunction of the Hecke operator T^p with eigenvalue λ_p prime to p. We describe a procedure for deriving a measure of Z_D.

Fix an integer N prime to p and to D. Let F be the D-module of all functions on rational numbers with denominator coprime to N. For any prime l not dividing N we define the Hecke operator T_l by

${\displaystyle (T_{l}f)\left({\frac {a}{b}}\right)=f\left({\frac {la}{b}}\right)+\sum _{k=0}^{l-1}f\left({\frac {a+kb}{lb}}\right)-\sum _{k=0}^{l-1}f\left({\frac {k}{l}}\right)\ .}$ 

Let f be an eigenfunction for T_p with eigenvalue λ_p in D. The quadratic equation X² − λ_pX + p = 0 has roots π₁, π₂ with π₁ a unit and π₂ divisible by p. Define a sequence a₀ = 2, a₁ = π₁+π₂ = λ_p and

${\displaystyle a_{k+2}=\lambda _{p}a_{k+1}-pa_{k}\ ,}$ 

so that

${\displaystyle a_{k}=\pi _{1}^{k}+\pi _{2}^{k}\ .}$ 

• Kubert, Daniel S.; Lang, Serge (1981). Modular Units. Grundlehren der Mathematischen Wissenschaften. Vol. 244. Springer-Verlag. ISBN 0-387-90517-0. Zbl 0492.12002.
• Lang, Serge (1990). Cyclotomic Fields I and II. Graduate Texts in Mathematics. Vol. 121 (second combined ed.). Springer Verlag. ISBN 3-540-96671-4. Zbl 0704.11038.
• Mazur, B.; Swinnerton-Dyer, P. (1974). "Arithmetic of Weil curves". Inventiones Mathematicae. 25: 1–61. doi:10.1007/BF01389997. Zbl 0281.14016.