Contracting prime sums, prime counts and densities along a fibre count #
Let K and E be number fields, T a set of height-one primes of ๐ E and S one of ๐ K,
and let ฯ send each prime of E to a prime of K. Suppose that ฯ maps T into S, that it
preserves absolute norms on T, and that every ๐ญ โ S has exactly c โ 0 preimages in T.
Then, for every real s,
โ_{๐ โ T} N๐ ^ (-s) = c * โ_{๐ญ โ S} N๐ญ ^ (-s),
and consequently T has Dirichlet density ฮด exactly when S has Dirichlet density ฮด / c.
The fibre count only matters away from a set of density zero. If ฯ does not increase norms and
has boundedly many preimages in T over each prime of S, the prime sum over T is bounded by a
multiple of the prime sum over S, so preimages of a density-zero set have density zero. Hence the
exact count c is needed only for the primes of S outside a density-zero set Z, with a
uniform bound over Z.
The typical ฯ is contraction ๐ โฆ ๐ โฉ ๐ K for an extension E / K. It preserves norms exactly
on the primes of residue degree one over K, and the others have density zero, so only primes of
residue degree one need to be counted in the fibres; at most [E : K] primes of E lie over a
given prime of K, which supplies the uniform bound over Z. This is how a density computed over
an extension field is transported down to the base, as in the proof of the Chebotarev density
theorem, where a relative Frobenius fibre over the fixed field of a cyclic subgroup is counted
over the primes of the base field.
The same fibre count multiplies the prime counts: for every cutoff x, ฯ_T(x) = c ฯ_S(x) and
ฯ_T(x) = c ฯ_S(x), with no error term and without the hypothesis c โ 0. By the prime ideal
theorem the ordinary prime counts of both fields are asymptotic to x / log x (and the weighted
counts ฯ to x), so the transfer, its bounded-fibre variant, and the contraction form hold for
natural density as well.
Main results #
NumberField.Set.primeIdealZetaSum_eq_mul_of_card_fiber: the exact identity of prime sums.NumberField.Set.hasDirichletDensity_iff_of_card_fiber: the transfer of Dirichlet densities.NumberField.Set.primeIdealZetaSum_le_mul_of_encard_fiber_le: the prime-sum inequality for a bounded fibre count.NumberField.Set.HasDirichletDensity.zero_of_encard_fiber_le: density zero pulls back along a bounded fibre count.NumberField.Set.hasDirichletDensity_iff_of_card_fiber_of_negligible: the transfer of Dirichlet densities when the fibre count is exact only off a set of density zero.NumberField.Set.hasDirichletDensity_contraction: the transfer along contraction, counting only primes of residue degree one and only off a set of density zero.NumberField.Set.primeCount_eq_mul_of_card_fiberandNumberField.Set.primeTheta_eq_mul_of_card_fiber: the exact identities of prime counts.NumberField.Set.primeCount_le_mul_of_encard_fiber_leandNumberField.Set.HasNaturalDensity.zero_of_encard_fiber_le: the bounded fibre count for prime counts and natural density zero.NumberField.Set.hasNaturalDensity_iff_of_card_fiber,NumberField.Set.hasNaturalDensity_iff_of_card_fiber_of_negligibleandNumberField.Set.hasNaturalDensity_contraction: the three transfers for natural density.
References #
- J. Neukirch, Algebraic Number Theory, Chapter VII, ยง13.
- J.-P. Serre, A Course in Arithmetic, Chapter VI, ยง4.1.
Prime sums along a fibre count. If ฯ maps T into S, preserves absolute norms on T,
and every prime of S has exactly c โ 0 preimages in T, then the prime sum over T is c
times the prime sum over S, at every real s.
Dirichlet densities along a fibre count. If ฯ maps T into S, preserves absolute
norms on T, and every prime of S has exactly c โ 0 preimages in T, then T has Dirichlet
density ฮด exactly when S has Dirichlet density ฮด / c.
Prime sums along a bounded fibre count. If ฯ maps T into S, does not increase
absolute norms on T, and every prime of S has at most m preimages in T, then for s โฅ 0 the
prime sum over T is at most m times the prime sum over S, provided the latter converges.
Density zero pulls back along a bounded fibre count. If S has Dirichlet density zero, and
ฯ maps T into S, does not increase absolute norms on T, and has at most m preimages in
T over each prime of S, then T has Dirichlet density zero.
Dirichlet densities along a fibre count off a negligible set. Let ฯ map the part of T
away from the preimage of Z into S \ Z, preserve absolute norms there, and not increase norms
over Z. Suppose that every prime of S outside the density-zero set Z has exactly c โ 0
preimages in T, and every prime in Z has at most m. Then T has Dirichlet density ฮด
exactly when S has Dirichlet density ฮด / c.
Dirichlet densities along contraction. Let E / K be an extension of number fields, T a
set of primes of E whose residue-degree-one members outside the preimage of Z contract into a
set S of primes of K, and Z a set of primes of K of Dirichlet density zero. If every prime
of S outside Z lies below exactly c โ 0 members of T of residue degree one over K, then
T has Dirichlet density ฮด exactly when S has Dirichlet density ฮด / c.
Neither the primes of T of residue degree above one over K nor those over Z need to be
counted: both form sets of density zero.
Prime counts and natural density along a fibre count #
Prime counts along a fibre count. If ฯ maps T into S, preserves absolute norms on
T, and every prime of S has exactly c preimages in T, then for every cutoff x there are
c times as many primes of T as primes of S of norm at most x.
Logarithmically weighted prime counts along a fibre count. If ฯ maps T into S,
preserves absolute norms on T, and every prime of S has exactly c preimages in T, then
ฯ_T(x) = c ฯ_S(x) for every cutoff x.
Natural densities along a fibre count. If ฯ maps T into S, preserves absolute norms
on T, and every prime of S has exactly c โ 0 preimages in T, then T has natural density
ฮด exactly when S has natural density ฮด / c.
Prime counts along a bounded fibre count. If ฯ maps T into S, does not increase
absolute norms on T, and every prime of S has at most m preimages in T, then for every
cutoff x there are at most m times as many primes of T as primes of S of norm at most
x.
Natural density zero pulls back along a bounded fibre count. If S has natural density
zero, and ฯ maps T into S, does not increase absolute norms on T, and has at most m
preimages in T over each prime of S, then T has natural density zero.
Natural densities along a fibre count off a negligible set. Let ฯ map the part of T
away from the preimage of Z into S \ Z, preserve absolute norms there, and not increase norms
over Z. Suppose that every prime of S outside the natural-density-zero set Z has exactly
c โ 0 preimages in T, and every prime in Z has at most m. Then T has natural density ฮด
exactly when S has natural density ฮด / c.
Natural densities along contraction. Let E / K be an extension of number fields, T a
set of primes of E whose residue-degree-one members outside the preimage of Z contract into a
set S of primes of K, and Z a set of primes of K of natural density zero. If every prime of
S outside Z lies below exactly c โ 0 members of T of residue degree one over K, then T
has natural density ฮด exactly when S has natural density ฮด / c.
As for NumberField.Set.hasDirichletDensity_contraction, neither the primes of T of residue
degree above one over K nor those over Z need to be counted.