The Riemann zeta function and tuning: Difference between revisions

Line 128:

\zeta(s) = \sum_n n^{-s}</math>

Now let's do two things: we're going to expand {{nowrap|''s'' {{=}} σ + ''it''}}, and we're going to multiply ζ(s) by its conjugate ζ(''s'')', noting that {{nowrap|ζ(''s'')' {{=}} ζ(''s''~~{{'}}~~)}} and {{nowrap|ζ(''s'') ⋅ ζ(''s'')' {{=}} ζ(''s'')<sup>2</sup>}}. We get:

Now let's do two things: we're going to expand {{nowrap|''s'' {{=}} σ + ''it''}}, and we're going to multiply ζ(s) by its conjugate ζ(''s'')′, noting that {{nowrap|ζ(''s'')' {{=}} ζ(''s''′)}} and {{nowrap|ζ(''s'') ⋅ ζ(''s'')' {{=}} ζ(''s'')<sup>2</sup>}}. We get:

<math> \displaystyle

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Note that since there's no restriction that n and d be coprime, the "rationals" we're using here don't have to be reduced. So this shows that zeta yields an error metric over all unreduced rationals, but leaves open the question of how reduced rationals are handled. It turns out that the same function also measures the error of reduced rationals, scaled only by a rolloff-dependent constant factor across all edos.

To see this, let's first note that every "unreduced" rational {{sfrac|''n''|''d''}} can be decomposed into the product of a reduced rational {{sfrac|''n''~~{{'}}~~|''d''{{-'}}}} and a common factor {{sfrac|''c''|''c''}}. Furthermore, note that for any reduced rational {{sfrac|''n''~~{{'}}~~|''d''{{-'}}}}, we can generate all unreduced rationals {{sfrac|''n''|''d''}} corresponding to it by multiplying it by all such common factors {{sfrac|''c''|''c''}}, where ''c'' is a strictly positive natural number.

To see this, let's first note that every "unreduced" rational {{sfrac|''n''|''d''}} can be decomposed into the product of a reduced rational {{sfrac|''n''′|''d''{{-'}}}} and a common factor {{sfrac|''c''|''c''}}. Furthermore, note that for any reduced rational {{sfrac|''n''′|''d''{{-'}}}}, we can generate all unreduced rationals {{sfrac|''n''|''d''}} corresponding to it by multiplying it by all such common factors {{sfrac|''c''|''c''}}, where ''c'' is a strictly positive natural number.

This allows us to change our original summation so that it's over three variables, ''n''~~{{'}}~~, ''d''~~{{-'}}~~, and ''c''~~{{'}}~~, where ''n''~~{{'}}~~ and ''d''~~{{-'}}~~ are coprime, and ''c'' is a strictly positive natural number:

This allows us to change our original summation so that it's over three variables, ''n''′, ''d''′, and ''c''′, where ''n''′ and ''d''′ are coprime, and ''c'' is a strictly positive natural number:

<math> \displaystyle

@@ Line 128: / Line 128: @@
 \zeta(s) = \sum_n n^{-s}</math>
-Now let's do two things: we're going to expand {{nowrap|''s'' {{=}} σ + ''it''}}, and we're going to multiply ζ(s) by its conjugate ζ(''s'')', noting that {{nowrap|ζ(''s'')' {{=}} ζ(''s''{{'}})}} and {{nowrap|ζ(''s'') ⋅ ζ(''s'')' {{=}} ζ(''s'')<sup>2</sup>}}. We get:
+Now let's do two things: we're going to expand {{nowrap|''s'' {{=}} σ + ''it''}}, and we're going to multiply ζ(s) by its conjugate ζ(''s'')′, noting that {{nowrap|ζ(''s'')' {{=}} ζ(''s''′)}} and {{nowrap|ζ(''s'') ⋅ ζ(''s'')' {{=}} ζ(''s'')<sup>2</sup>}}. We get:
 <math> \displaystyle
@@ Line 242: / Line 242: @@
 Note that since there's no restriction that n and d be coprime, the "rationals" we're using here don't have to be reduced. So this shows that zeta yields an error metric over all unreduced rationals, but leaves open the question of how reduced rationals are handled. It turns out that the same function also measures the error of reduced rationals, scaled only by a rolloff-dependent constant factor across all edos.
-To see this, let's first note that every "unreduced" rational {{sfrac|''n''|''d''}} can be decomposed into the product of a reduced rational {{sfrac|''n''{{'}}|''d''{{-'}}}} and a common factor {{sfrac|''c''|''c''}}. Furthermore, note that for any reduced rational {{sfrac|''n''{{'}}|''d''{{-'}}}}, we can generate all unreduced rationals {{sfrac|''n''|''d''}} corresponding to it by multiplying it by all such common factors {{sfrac|''c''|''c''}}, where ''c'' is a strictly positive natural number.
+To see this, let's first note that every "unreduced" rational {{sfrac|''n''|''d''}} can be decomposed into the product of a reduced rational {{sfrac|''n''′|''d''{{-'}}}} and a common factor {{sfrac|''c''|''c''}}. Furthermore, note that for any reduced rational {{sfrac|''n''′|''d''{{-'}}}}, we can generate all unreduced rationals {{sfrac|''n''|''d''}} corresponding to it by multiplying it by all such common factors {{sfrac|''c''|''c''}}, where ''c'' is a strictly positive natural number.
-This allows us to change our original summation so that it's over three variables, ''n''{{'}}, ''d''{{-'}}, and ''c''{{'}}, where ''n''{{'}} and ''d''{{-'}} are coprime, and ''c'' is a strictly positive natural number:
+This allows us to change our original summation so that it's over three variables, ''n''′, ''d''′, and ''c''′, where ''n''′ and ''d''′ are coprime, and ''c'' is a strictly positive natural number:
 <math> \displaystyle