The Riemann zeta function and tuning: Difference between revisions

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Note that since there is no restriction that ''n'' and ''d'' be coprime, the rationals we are using here do not 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 us 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 us 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

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==== Zeta peak integer edos ====

Alternatively (as [[groundfault]] has found), if we do not allow octave detuning and instead look at only the record {{nowrap|{{abs|Z(''x'')}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs| 1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}} … of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}} … and edos present in the previous list but not present here include {{EDOs| 4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}} … with 72's removal perhaps being the most surprising, showing ~~the strength of 53 in~~ that 72 does not improve on 53's peak. This definition may be better for measuring how accurate edos are without detuned octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos."

Alternatively (as [[groundfault]] has found), if we do not allow octave detuning and instead look at only the record {{nowrap|{{abs|Z(''x'')}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs| 1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}} … of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}} … and edos present in the previous list but not present here include {{EDOs| 4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}} … with 72's removal perhaps being the most surprising, showing that pure-octave 72 does not improve on 53's peak while stretched 72 does. This definition may be better for measuring how accurate edos are without detuned octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos."

==== Zeta integral edos ====

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=== Anti-record edos ===

==== Zeta valley edos ====

Just like with zeta peak edos which have progressively higher {{nowrap|{{abs|Z(x)}}}} scores, we can also look at edos with progressively ''lower'' {{nowrap|{{abs|Z(x)}}}} for integer values of ''x''. ~~These correspond to ''zeta valley edos'', and we get~~ {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}}… Zeta valley edos can be thought of as pure-octave tunings that tend to deviate from ''p''-limit JI as much as possible while still preserving octaves, and can serve as "more xenharmonic" tunings. Zeta valley edos are only measured with pure octaves, since "tempered-octave zeta valley edos" would simply be any zero of Z(x). Keep in mind, however, that the ''most'' xenharmonic tunings would not contain octaves at all.

Just like with zeta peak edos which have progressively higher {{nowrap|{{abs|Z(x)}}}} scores, we can also look at edos with progressively ''lower'' {{nowrap|{{abs|Z(x)}}}} for integer values of ''x''. This gives us {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}}… which correspond to ''zeta valley edos''. Zeta valley edos can be thought of as pure-octave tunings that tend to deviate from ''p''-limit JI as much as possible while still preserving octaves, and can serve as "more xenharmonic" tunings. Zeta valley edos are only measured with pure octaves, since "tempered-octave zeta valley edos" would simply be any zero of Z(x). Keep in mind, however, that the ''most'' xenharmonic tunings (essentially, tuning systems that avoid ''all'' ''p''-limit JI as much as possible) would not contain octaves at all.

Notice that there is a very large jump from [[79edo]] to [[5941edo]]. We know that record {{nowrap|{{abs|Z(x)}}}} scores, both with tempered octaves and pure octaves, grow logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval {{nowrap|[0, ''c'' log(''x'')]}}, the probability for the next edo to have a zeta score less than a given small value is also very small, so we would expect valley edos to be rarer than peak edos. So, it would be more productive to find edos which zeta score is simply less than a given threshold.

=== Absolute zeta peak edos ===

=== Other lists ===

{{Idiosyncratic terms|"Absolute zeta peak edos" was coined by {{u|Godtone}}, "''k''-ary-peak edos" was coined by {{u|Akselai}}, the types of "local zeta" edos and "indecisive edos" were coined by [[Budjarn Lambeth]].}}

==== Absolute zeta peak edos ====

If we consider that zeta is a measure of relative error (that is, error measured relative to the step size), we realize that plenty of equal temperaments<ref group="note">Note importantly that we speak of ''equal temperaments'' rather than ''edos'' because generally a record peak ''does not'' correspond to an edo, which can have tangible consequences (a significant example is discussed in the next section).</ref> are excluded simply because, though practically speaking they have great tuning properties, they are not as "efficient" with their number of tones as the last record peak. Arguably what we are interested in is a sequence of edos that generally do increasingly better at tuning JI in terms of lowering the average cent error. Therefore, it suffices to multiply the score by the size of the equal temperament. Surprisingly, the list for ''s'' = 1/2 — which is supposedly where high-limit information is maximized — is ''almost identical'' to the one for ''s'' = 1 — which is the smallest value of ''s'' that we can assume to be meaningful without assuming that the analytic continuation preserves the tuning properties we are interested in — so that we have reassurance from the ''s'' = 1 list that the ''s'' = 1/2 list is meaningful wherever they agree. This is important because surprisingly, the two lists of equal temperament are ''identical up to [[311edo|311et]]'', with only one edo, [[8edo]], omitted from the list for {{nowrap| ''s'' {{=}} 1 }}. This list is {{EDOs| 1, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15, 17, 19, 22, 24, 26, 27, 31, 34, 41, 46, 53, 58, 65, 68, 72, 84, 87, 94, 99, 111, 118, 130, 140, 152, 171, 183, 198, 212, 217, 224, 243, 270, 311, … }}.

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The equal temperaments added relative to the non-extended list of only things that are records proper are: {{EDOs| 6, 8, 11, 13, 16, 21, 29, 36, 38, 39, 43, 45, 48, 50, 56, 60, 63, 77, 80, 89, 96, 103, 106, 113, 121, 125, 137, 145, 149, 159, 161, 166, (176,) 190, 193, (202,) 229, 239, 248, 255, 277, 282, 289, 301 }}<ref group="note">39et is a notable example because 39edo corresponds to a zeta valley, so it is surprising that it would be included here; the reason that it is included is because this is ''not'' 39edo, but 39 ''equal temperament'', corresponding to a 3.8{{cent}} flat-tempered octave so that it is actually ~39.124edo, that is, it corresponds to the 173rd zeta peak, known by the shorthand 173zpi (where i stands for index). Therefore, this may prove a good testcase for investigating the effects of zeta-based octave-tempering, though given the size of the stretch, the difference is likely to be subtle, but the fact that it "changes zeta's mind" this much is itself interesting. You can also interpret this result differently, which is as evidence that you should not include equal temperaments worse than the third-best-scoring equal temperament so far, given the somewhat dubious inclusion of 39et, however it should be noted that this is more to do with that at the very beginning of the list there are not many equal temperaments to "beat" so that beating the third-best-scoring equal temperament so far is easy, though arguably this is not a flaw because people are often more likely to try a smaller equal temperament. It is also perhaps worth noting that 37et almost makes this extended list, but the omission of 37et is much better addressed by no-3's zeta.</ref>.

=== ''k''-ary-peak edos ===

==== ''k''-ary-peak edos ====

~~{{Idiosyncratic terms|the term "''k''-ary-peak edos" itself, as well as the names for the different types of ''k''-ary-peak edos. Used by {{u|Akselai}} and [[Budjarn Lambeth]].}}~~

If we want to find the second-best edos ranked by zeta peaks, then given a full list of zeta peaks, we can remove the successively higher peaks to get another sequence of succesively higher peaks, which correspond to edos called '''2-ary peak edos''': defined as non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak edo.

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We can repeat this process as many times as we want, resulting in '''''k''-ary-peak edos'''. The ordinary peak edos are 1-ary peak edos, then there are 2-ary peak edos, 3-ary peak edos, and so on. However keep in mind that the higher ''k'' gets, the less meaningful the peaks will get, especially for smaller edos (less than about 100).

~~=== Non-record edos ===~~

~~{{Idiosyncratic terms|the names for the different types of non-record edos. Proposed by [[Budjarn Lambeth]]}}~~

~~The following lists of edos are not determined by successively large measured values, they are edos that satisfy some other property relating to zeta peaks instead.~~

==== Local zeta peak edos ====

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Removing 2 leads to increasing adjusted peak values corresponding to edts into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 75, 88, 131, 245, 316,…}} parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen–Pierce]] division of the tritave, but the multiples 26 and 39 also.

~~== Open problems ==~~

~~# Are there metrics similar to zeta metrics, but for edos' performance at approximating arbitrary [[delta-rational]] chords?~~

== Further information ==

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[[Category:Number theory]]

[[Category:Pages with proofs]]

~~[[Category:Pages with open problems]]~~

@@ Line 266: / Line 266: @@
 Note that since there is no restriction that ''n'' and ''d'' be coprime, the rationals we are using here do not 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 us 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 us 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 346: / Line 346: @@
 ==== Zeta peak integer edos ====
-Alternatively (as [[groundfault]] has found), if we do not allow octave detuning and instead look at only the record {{nowrap|{{abs|Z(''x'')}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs| 1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}} … of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}} … and edos present in the previous list but not present here include {{EDOs| 4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}} … with 72's removal perhaps being the most surprising, showing the strength of 53 in that 72 does not improve on 53's peak. This definition may be better for measuring how accurate edos are without detuned octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos."
+Alternatively (as [[groundfault]] has found), if we do not allow octave detuning and instead look at only the record {{nowrap|{{abs|Z(''x'')}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs| 1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}} … of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}} … and edos present in the previous list but not present here include {{EDOs| 4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}} … with 72's removal perhaps being the most surprising, showing that pure-octave 72 does not improve on 53's peak while stretched 72 does. This definition may be better for measuring how accurate edos are without detuned octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos."
 ==== Zeta integral edos ====
@@ Line 551: / Line 551: @@
 === Anti-record edos ===
 ==== Zeta valley edos ====
-Just like with zeta peak edos which have progressively higher {{nowrap|{{abs|Z(x)}}}} scores, we can also look at edos with progressively ''lower'' {{nowrap|{{abs|Z(x)}}}} for integer values of ''x''. These correspond to ''zeta valley edos'', and we get {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}}… Zeta valley edos can be thought of as pure-octave tunings that tend to deviate from ''p''-limit JI as much as possible while still preserving octaves, and can serve as "more xenharmonic" tunings. Zeta valley edos are only measured with pure octaves, since "tempered-octave zeta valley edos" would simply be any zero of Z(x). Keep in mind, however, that the ''most'' xenharmonic tunings would not contain octaves at all.
+Just like with zeta peak edos which have progressively higher {{nowrap|{{abs|Z(x)}}}} scores, we can also look at edos with progressively ''lower'' {{nowrap|{{abs|Z(x)}}}} for integer values of ''x''. This gives us {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}}… which correspond to ''zeta valley edos''. Zeta valley edos can be thought of as pure-octave tunings that tend to deviate from ''p''-limit JI as much as possible while still preserving octaves, and can serve as "more xenharmonic" tunings. Zeta valley edos are only measured with pure octaves, since "tempered-octave zeta valley edos" would simply be any zero of Z(x). Keep in mind, however, that the ''most'' xenharmonic tunings (essentially, tuning systems that avoid ''all'' ''p''-limit JI as much as possible) would not contain octaves at all.
 Notice that there is a very large jump from [[79edo]] to [[5941edo]]. We know that record {{nowrap|{{abs|Z(x)}}}} scores, both with tempered octaves and pure octaves, grow logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval {{nowrap|[0, ''c'' log(''x'')]}}, the probability for the next edo to have a zeta score less than a given small value is also very small, so we would expect valley edos to be rarer than peak edos. So, it would be more productive to find edos which zeta score is simply less than a given threshold.
-=== Absolute zeta peak edos ===
+=== Other lists ===
+{{Idiosyncratic terms|"Absolute zeta peak edos" was coined by {{u|Godtone}}, "''k''-ary-peak edos" was coined by {{u|Akselai}}, the types of "local zeta" edos and "indecisive edos" were coined by [[Budjarn Lambeth]].}}
+==== Absolute zeta peak edos ====
 If we consider that zeta is a measure of relative error (that is, error measured relative to the step size), we realize that plenty of equal temperaments<ref group="note">Note importantly that we speak of ''equal temperaments'' rather than ''edos'' because generally a record peak ''does not'' correspond to an edo, which can have tangible consequences (a significant example is discussed in the next section).</ref> are excluded simply because, though practically speaking they have great tuning properties, they are not as "efficient" with their number of tones as the last record peak. Arguably what we are interested in is a sequence of edos that generally do increasingly better at tuning JI in terms of lowering the average cent error. Therefore, it suffices to multiply the score by the size of the equal temperament. Surprisingly, the list for ''s'' = 1/2 — which is supposedly where high-limit information is maximized — is ''almost identical'' to the one for ''s'' = 1 — which is the smallest value of ''s'' that we can assume to be meaningful without assuming that the analytic continuation preserves the tuning properties we are interested in — so that we have reassurance from the ''s'' = 1 list that the ''s'' = 1/2 list is meaningful wherever they agree. This is important because surprisingly, the two lists of equal temperament are ''identical up to [[311edo|311et]]'', with only one edo, [[8edo]], omitted from the list for {{nowrap| ''s'' {{=}} 1 }}. This list is {{EDOs| 1, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15, 17, 19, 22, 24, 26, 27, 31, 34, 41, 46, 53, 58, 65, 68, 72, 84, 87, 94, 99, 111, 118, 130, 140, 152, 171, 183, 198, 212, 217, 224, 243, 270, 311, … }}.
@@ Line 563: / Line 566: @@
 The equal temperaments added relative to the non-extended list of only things that are records proper are: {{EDOs| 6, 8, 11, 13, 16, 21, 29, 36, 38, 39, 43, 45, 48, 50, 56, 60, 63, 77, 80, 89, 96, 103, 106, 113, 121, 125, 137, 145, 149, 159, 161, 166, (176,) 190, 193, (202,) 229, 239, 248, 255, 277, 282, 289, 301 }}<ref group="note">39et is a notable example because 39edo corresponds to a zeta valley, so it is surprising that it would be included here; the reason that it is included is because this is ''not'' 39edo, but 39 ''equal temperament'', corresponding to a 3.8{{cent}} flat-tempered octave so that it is actually ~39.124edo, that is, it corresponds to the 173rd zeta peak, known by the shorthand 173zpi (where i stands for index). Therefore, this may prove a good testcase for investigating the effects of zeta-based octave-tempering, though given the size of the stretch, the difference is likely to be subtle, but the fact that it "changes zeta's mind" this much is itself interesting. You can also interpret this result differently, which is as evidence that you should not include equal temperaments worse than the third-best-scoring equal temperament so far, given the somewhat dubious inclusion of 39et, however it should be noted that this is more to do with that at the very beginning of the list there are not many equal temperaments to "beat" so that beating the third-best-scoring equal temperament so far is easy, though arguably this is not a flaw because people are often more likely to try a smaller equal temperament. It is also perhaps worth noting that 37et almost makes this extended list, but the omission of 37et is much better addressed by no-3's zeta.</ref>.
-=== ''k''-ary-peak edos ===
+==== ''k''-ary-peak edos ====
-{{Idiosyncratic terms|the term "''k''-ary-peak edos" itself, as well as the names for the different types of ''k''-ary-peak edos. Used by {{u|Akselai}} and [[Budjarn Lambeth]].}}
 If we want to find the second-best edos ranked by zeta peaks, then given a full list of zeta peaks, we can remove the successively higher peaks to get another sequence of succesively higher peaks, which correspond to edos called '''2-ary peak edos''': defined as non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak edo.
@@ Line 576: / Line 577: @@
 We can repeat this process as many times as we want, resulting in '''''k''-ary-peak edos'''. The ordinary peak edos are 1-ary peak edos, then there are 2-ary peak edos, 3-ary peak edos, and so on. However keep in mind that the higher ''k'' gets, the less meaningful the peaks will get, especially for smaller edos (less than about 100).
-=== Non-record edos ===
-{{Idiosyncratic terms|the names for the different types of non-record edos. Proposed by [[Budjarn Lambeth]]}}
-The following lists of edos are not determined by successively large measured values, they are edos that satisfy some other property relating to zeta peaks instead.
 ==== Local zeta peak edos ====
@@ Line 649: / Line 645: @@
 Removing 2 leads to increasing adjusted peak values corresponding to edts into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 75, 88, 131, 245, 316,…}} parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen–Pierce]] division of the tritave, but the multiples 26 and 39 also.
-== Open problems ==
-# Are there metrics similar to zeta metrics, but for edos' performance at approximating arbitrary [[delta-rational]] chords?
 == Further information ==
@@ Line 674: / Line 667: @@
 [[Category:Number theory]]
 [[Category:Pages with proofs]]
-[[Category:Pages with open problems]]
 {{Todo| increase applicability | simplify }}