dirichlet.spad

Martin Rubey

July 14, 2013

Abstract

The domain defined in this file models the Dirichlet ring,

Contents

1 The Dirichlet Ring 1

2 domain DIRRING DirichletRing 1

3 License 5

1 The Dirichlet Ring

The Dirichlet Ring is the ring of arithmetical functions

f : ℕ_{+} → R

(see http://en.wikipedia.org/wiki/Arithmetic_function) together with the
Dirichlet convolution (see http://en.wikipedia.org/wiki/Dirichlet_convolution)

as multiplication and component-wise addition. Since we can consider the values an arithmetic functions assumes as the coefficients of a Dirichlet generating series, we call R the coefficient ring of a function.

In general we only assume that the coefficient ring R is a ring. If R happens to be commutative, then so is the Dirichlet ring, and in this case it is even an algebra.

Apart from the operations inherited from those categories, we only provide some convenient coercion functions.

2 domain DIRRING DirichletRing

⟨domain DIRRING DirichletRing⟩≡

  )abbrev domain DIRRING DirichletRing
  ++ Author: Martin Rubey
  ++ Description: DirichletRing is the ring of arithmetical functions

++ with Dirichlet convolution as multiplication
DirichletRing(Coef: Ring):
    Exports == Implementation where
PI ==> PositiveInteger
FUN ==> PI -> Coef
Exports ==> Join(Ring, Eltable(PI, Coef)) with
if Coef has CommutativeRing then
    IntegralDomain
if Coef has CommutativeRing then
    Algebra Coef
coerce: FUN -> %
coerce: % -> FUN
coerce: Stream Coef -> %
coerce: % -> Stream Coef
zeta: constant -> %
++ zeta() returns the function which is constantly one
multiplicative?: % -> Boolean
++ multiplicative?(a) returns true if the first
++ $streamCount$Lisp coefficients of a are multiplicative
additive?: % -> Boolean
++ additive?(a) returns true if the first
++ $streamCount$Lisp coefficients of a are additive
Implementation ==> add
Rep := Record(function: FUN)
per(f: Rep): % == f pretend %
rep(a: %): Rep == a pretend Rep
elt(a: %, n: PI): Coef ==
    f: FUN := (rep a).function
    f n
coerce(a: %): FUN == (rep a).function
coerce(f: FUN): % == per [f]

indices: Stream Integer
        := integers(1)$StreamTaylorSeriesOperations(Integer)
coerce(a: %): Stream Coef ==
    f: FUN := (rep a).function
    map((n: Integer): Coef +-> f(n::PI), indices)
       $StreamFunctions2(Integer, Coef)
coerce(f: Stream Coef): % ==
    ((n: PI): Coef +-> f.(n::Integer))::%
coerce(f: %): OutputForm == f::Stream Coef::OutputForm
1: % ==
    ((n: PI): Coef +-> (if one? n then 1$Coef else 0$Coef))::%
0: % ==
    ((n: PI): Coef +-> 0$Coef)::%
zeta: % ==
    ((n: PI): Coef +-> 1$Coef)::%
(f: %) + (g: %) ==
    ((n: PI): Coef +-> f(n)+g(n))::%
- (f: %) ==
    ((n: PI): Coef +-> -f(n))::%
(a: Integer) * (f: %) ==
    ((n: PI): Coef +-> a*f(n))::%
(a: Coef) * (f: %) ==
    ((n: PI): Coef +-> a*f(n))::%
import IntegerNumberTheoryFunctions
(f: %) * (g: %) ==
    conv := (n: PI): Coef +-> _
         reduce((a: Coef, b: Coef): Coef +-> a + b, _
                [f(d::PI) * g((n quo d)::PI) for d in divisors(n::Integer)], 0)
                $ListFunctions2(Coef, Coef)
    conv::%
unit?(a: %): Boolean == not (recip(a(1$PI))$Coef case "failed")
qrecip: (%, Coef, PI) -> Coef

qrecip(f: %, f1inv: Coef, n: PI): Coef ==
if one? n then f1inv
else
-f1inv * reduce(_+, [f(d::PI) * qrecip(f, f1inv, (n quo d)::PI) _
                     for d in rest divisors(n)], 0) _
               $ListFunctions2(Coef, Coef)
recip f ==
if (f1inv := recip(f(1$PI))$Coef) case "failed" then "failed"
else
mp := (n: PI): Coef +-> qrecip(f, f1inv, n)
mp::%::Union(%, "failed")
multiplicative? a ==
n: Integer := _$streamCount$Lisp
for i in 2..n repeat
fl := factors(factor i)$Factored(Integer)
rl := [a.(((f.factor)::PI)^((f.exponent)::PI)) for f in fl]
if a.(i::PI) ~= reduce((r:Coef, s:Coef):Coef +-> r*s, rl)
then
    output(i::OutputForm)$OutputPackage
    output(rl::OutputForm)$OutputPackage
    return false
true
additive? a ==
n: Integer := _$streamCount$Lisp
for i in 2..n repeat
fl := factors(factor i)$Factored(Integer)
rl := [a.(((f.factor)::PI)^((f.exponent)::PI)) for f in fl]
if a.(i::PI) ~= reduce((r:Coef, s:Coef):Coef +-> r+s, rl)
then
    output(i::OutputForm)$OutputPackage
    output(rl::OutputForm)$OutputPackage
    return false
true

3 License

⟨license⟩≡

--Copyright (c) 2010, Martin Rubey <Martin.Rubey@math.uni-hannover.de>
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
--    - Redistributions of source code must retain the above copyright
--      notice, this list of conditions and the following disclaimer.
--
--    - Redistributions in binary form must reproduce the above copyright
--      notice, this list of conditions and the following disclaimer in
--      the documentation and/or other materials provided with the
--      distribution.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

⟨* ⟩≡ ⟨license⟩

⟨domain DIRRING DirichletRing⟩