The zeros and the logarithm of an ideal Euler product #
A TauCeti.EulerProductData over a number field K carries a coprime-multiplicative coefficient
function on the nonzero ideals of ๐ K, together with its prime-power local data, and where its
ideal-indexed Dirichlet series converges absolutely its local factors
TauCeti.EulerProductData.eulerFactor have an unrestricted product equal to the L-series of its
norm coefficients
(TauCeti.EulerProductData.hasProd_eulerFactor). A convergent infinite product of nonzero factors
may still vanish, so that identity alone decides neither whether the L-series vanishes nor whether
the product has a logarithm.
This file settles both questions for a general Euler product. The local factor at P is
1 + โ_{e โฅ 1} D(P ^ e) N(P) ^ (-e s), and the tails over the distinct prime-power ideals form a
subfamily of the absolutely convergent ideal-indexed series; so the local factors are 1 up to a
summable error. For such a product the two questions have the same answer: the L-series vanishes
exactly when some local factor does, and only finitely many local factors can vanish at all. When
none does, the sum of the principal logarithms of the local factors converges and its exponential is
the L-series.
The degree-one case is separate: a completely multiplicative weight has local factors
(1 - ฯ(P) N(P) ^ (-s))โปยน, which are visibly nonzero, so there absolute convergence alone gives
nonvanishing (TauCeti.MultiplicativeIdealWeight.LSeries_ne_zero_of_summable_idealTerm). For
general data the hypothesis cannot be dropped: the local series is an arbitrary power series in
N(P) ^ (-s) with constant term 1, and already a local factor 1 + c N(P) ^ (-s) with c โ 0
vanishes for some s.
Main results #
TauCeti.EulerProductData.summable_eulerFactor_sub_one: the local Euler factors differ from1by a summable error.TauCeti.EulerProductData.norm_eulerFactor_sub_one_le_tsum_norm_of_re_le_re: the deviation of a local factor from1is uniformly bounded on a right half-plane by a prime-power tail.TauCeti.EulerProductData.eulerFactor_ne_zero_of_tsum_norm_lt_oneandTauCeti.EulerProductData.finite_setOf_eulerFactor_eq_zero: a local factor whose prime-power tail has norm sum less than1is nonzero, and only finitely many local factors vanish.TauCeti.EulerProductData.LSeries_eq_zero_iff_exists_eulerFactor_eq_zero: theL-series of the norm coefficients vanishes exactly when some local Euler factor vanishes.TauCeti.EulerProductData.exp_tsum_log_eulerFactor_eq_LSeries: with every local factor nonzero, theL-series is the exponential of the sum of the principal logarithms of the local factors.TauCeti.EulerProductData.exists_differentiableOn_exp_eq_LSeries: a holomorphic logarithm of theL-series on a simply connected region of absolute convergence with nonvanishing local factors.
References #
- J. Neukirch, Algebraic Number Theory, Chapter VII.
The local factors are 1 up to a summable error #
The local Euler factor, with its constant term split off. The e = 0 term of the local
series at P is the value at the unit ideal, which coprime multiplicativity fixes to be 1; only
convergence of the local series at P is needed.
The local Euler factors differ from 1 by a summable error. This is the quantitative
content of absolute convergence at s: the deviation of the local factor at P from 1 is the
tail of the local series, and those tails are a subfamily of the ideal-indexed series.
The deviation of a local Euler factor from 1 is at most the norm sum of the prime-power
tail at that prime.
The local Euler factors approach 1 uniformly on a half-plane of absolute convergence.
To the right of a point w at which the local series at P converges absolutely, the deviation of
the local factor at P from 1 is bounded, independently of the point, by the prime-power tail at
P computed at w.
Nonvanishing #
A local Euler factor with a small prime-power tail is nonzero. This is the criterion the
nonvanishing theorems below consume; it is checkable from a bound on the coefficients at the
powers of the single prime P.
Only finitely many local Euler factors can vanish. The deviations from 1 are summable,
hence smaller than 1 outside a finite set of primes.
A vanishing local Euler factor kills the L-series. Every finite partial product over a
set of primes containing the offending one is zero, and those partial products converge to the
L-series.
The Euler product of nonvanishing local factors does not vanish. Absolute convergence
makes the local factors 1 up to a summable error, and such a product vanishes only if one of its
factors does.
The L-series of a general Euler product vanishes exactly where a local factor does. On
the region of absolute convergence the zeros of an ideal Euler product are therefore local data.
The logarithm of the product #
The principal logarithms of the local Euler factors are summable. No nonvanishing
hypothesis is needed: Complex.log 0 = 0, and the finitely many vanishing factors of
TauCeti.EulerProductData.finite_setOf_eulerFactor_eq_zero cannot affect summability.
The Euler product in exponential form. Where the ideal-indexed Dirichlet series converges
absolutely and no local Euler factor vanishes, the L-series of the norm coefficients is the
exponential of the sum of the principal logarithms of the local factors.
As exp is not injective this does not exhibit a logarithm of the L-series; for that see
TauCeti.EulerProductData.exists_differentiableOn_exp_eq_LSeries.
A holomorphic logarithm of the L-series of an Euler product. On a simply connected open
set where the ideal-indexed Dirichlet series converges absolutely and no local Euler factor
vanishes, there is a holomorphic L with exp โ L the L-series of the norm coefficients, and
deriv L is its logarithmic derivative.
For a completely multiplicative weight the local hypothesis is automatic; in general it is the
hypothesis that cuts out a zero-free region, by
TauCeti.EulerProductData.LSeries_eq_zero_iff_exists_eulerFactor_eq_zero.