Deleting finitely many Euler factors #
Restricting Euler-product data away from a finite set S of primes, keeping only the
coefficients of the ideals prime to S, replaces the local Euler factors at S by 1 and leaves
the others untouched. On the half-plane of absolute convergence the two L-series therefore
differ by the finitely many deleted factors; for a completely multiplicative weight Ο the
restriction Ο.restrict S divides the L-series by β π β S, (1 - Ο(π) N(π) ^ (-s))β»ΒΉ.
For the trivial weight the restriction is ofBadPrimes S, the indicator of the ideals prime to
S, and its L-series is the Dedekind zeta function with the Euler factors at S removed:
L_S(s) = ΞΆ_K(s) * β π β S, (1 - N(π) ^ (-s)) for Re s > 1.
The correction factor does not vanish on Re s > 0, because
|N(π) ^ s| = N(π) ^ (Re s) > 1 there. As s β 1βΊ, the normalized expression
(s - 1) L_S(s) tends to dedekindZeta_residue K multiplied by the nonzero number
β π β S, (1 - N(π)β»ΒΉ). The logarithmic derivative of L_S differs from that of ΞΆ_K by the
finite sum β π β S, log N(π) / (N(π) ^ s - 1), which is holomorphic on Re s > 0 and in
particular across the line Re s = 1. This is the form in which Dirichlet series whose Euler
products omit the ramified primes, such as the trivial Galois-character series, are compared with
ΞΆ_K.
Main results #
TauCeti.EulerProductData.eulerFactor_restrictAway_of_mem,TauCeti.EulerProductData.eulerFactor_restrictAway_of_notMem: restricting away fromSreplaces the local factors atSby1and keeps the others.TauCeti.EulerProductData.LSeries_restrictAway_mul_prod_eulerFactor: multiplying theL-series of the restriction by the deleted local factors recovers the originalL-series.TauCeti.MultiplicativeIdealWeight.LSeries_restrict: the same for a completely multiplicative weight, with the deleted factors in closed form.TauCeti.LSeries_ofBadPrimes: theL-series of the indicator of the ideals prime toSisΞΆ_K(s) * β π β S, (1 - N(π) ^ (-s))onRe s > 1.TauCeti.prod_one_sub_absNorm_cpow_neg_ne_zero: the correction factor has no zero onRe s > 0.TauCeti.dedekindZeta_residue_mul_prod_one_sub_absNorm_cpow_neg_one_ne_zero: the corrected residue ats = 1is nonzero.TauCeti.tendsto_sub_one_mul_LSeries_ofBadPrimes: the normalized right-hand limit ats = 1.TauCeti.logDeriv_LSeries_ofBadPrimes: the logarithmic derivative onRe s > 1, andTauCeti.differentiableOn_sum_log_absNorm_div_cpow_sub_one: the correction term in it is holomorphic onRe s > 0.TauCeti.MultiplicativeIdealWeight.IsNormTwistOnGood.LSeries_normCoeffandTauCeti.MultiplicativeIdealWeight.IsNormTwistOnGood.tendsto_sub_one_mul_LSeries: a weight that is a norm twist with parameteruon its good ideals has forL-series such a deleted zeta function read ats - u * I, with the corresponding pole ats = 1 + u * I.
References #
- J. Neukirch, Algebraic Number Theory, Chapter VII.
Restricting away from a set of primes preserves absolute convergence of the ideal-indexed
Dirichlet series, since it only replaces some terms by 0.
Restricting away from S replaces the local Euler factor at a prime of S by 1.
Restricting away from S leaves the local Euler factor at a prime outside S unchanged.
Deleting finitely many Euler factors. Where the ideal-indexed Dirichlet series of D
converges absolutely, restricting D away from a finite set S of primes divides its L-series
by the local Euler factors at S: multiplying them back recovers the L-series of D.
Deleting finitely many Euler factors of a completely multiplicative weight. Where the
ideal-indexed Dirichlet series of Ο converges absolutely, restricting Ο away from a finite set
S of primes multiplies its L-series by β π β S, (1 - Ο(π) N(π) ^ (-s)), the reciprocal of the
deleted local factors.
The Dedekind zeta function with finitely many Euler factors deleted #
The deleted Euler-factor correction does not vanish on Re s > 0. In particular, its
value at s = 1 is nonzero.
The residue of the Dedekind zeta function remains nonzero after deleting finitely many
Euler factors at s = 1.
The Dedekind zeta function with the Euler factors at S deleted. For a finite set S of
primes and Re s > 1, the L-series of the indicator of the ideals prime to S is
ΞΆ_K(s) * β π β S, (1 - N(π) ^ (-s)).
The normalized right-hand limit at s = 1 after deleting Euler factors. As s β 1βΊ,
(s - 1) L_S(s) tends to dedekindZeta_residue K multiplied by
β π β S, (1 - N(π) ^ (-1)), which is nonzero by
prod_one_sub_absNorm_cpow_neg_ne_zero.
The logarithmic derivative after deleting Euler factors. For a finite set S of primes
and Re s > 1, the logarithmic derivative of L_S(s) = ΞΆ_K(s) * β π β S, (1 - N(π) ^ (-s)) is
that of ΞΆ_K plus β π β S, log N(π) / (N(π) ^ s - 1).
The Euler-factor correction is holomorphic on Re s > 0. The finite sum
β π β S, log N(π) / (N(π) ^ s - 1) by which logDeriv_LSeries_ofBadPrimes separates the
logarithmic derivatives of L_S and ΞΆ_K extends holomorphically across the line Re s = 1.
Weights that are norm twists on their good ideals #
The L-series of a weight that is a norm twist on its good ideals. Such a weight is the
twist by N(I) ^ (u * I) of the indicator of the ideals prime to its bad primes S, so its
L-series is the Dedekind zeta function with the Euler factors at S deleted, read at the
imaginary translate s - u * I.
The pole of the L-series of a norm twist on the good ideals. Along the horizontal ray
s = t + u * I with t β 1βΊ, the normalized series tends to the residue of the Dedekind zeta
function times the deleted Euler factors at s = 1; that limit is nonzero by
TauCeti.prod_one_sub_absNorm_cpow_neg_ne_zero and
NumberField.dedekindZeta_residue_pos.