Riemann zeta function: Difference between revisions

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Seeing that we notate the power as ''s'', it might become apparent where the Riemann zeta function will eventually show up.

If ''s'' is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting cancels this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of {{sfrac|1|''n''}} for each prime power ''p''''n''. A somewhat peculiar but useful way to write the result of doing this is in terms of the {{w|~~Von~~ Mangoldt function}}, an {{w|arithmetic function}} on positive integers which is equal to ln(''p'') on prime powers ''p''''n'', and is zero elsewhere. This is written using a capital lambda, as Λ(''n''), and in terms of it we can include prime powers in our error function as

If ''s'' is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting cancels this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of {{sfrac|1|''n''}} for each prime power ''p''''n''. A somewhat peculiar but useful way to write the result of doing this is in terms of the {{w|von Mangoldt function}}, an {{w|arithmetic function}} on positive integers which is equal to ln(''p'') on prime powers ''p''''n'', and is zero elsewhere. This is written using a capital lambda, as Λ(''n''), and in terms of it we can include prime powers in our error function as

<math>\displaystyle \xi_\infty(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\lfloor x \log_2 n \rceil^2}{n^s}</math>

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This new function has the property that {{nowrap|Fs(''x'') {{=}} Fs(0) − Es(''x'')}}, so that all we have done is flip the sign of Es(''x'') and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum.

Of more interest is the fact that it is a known mathematical function.

The logarithm of the {{w|Riemann zeta function}} function can be expressed in terms of a {{w|Dirichlet series}} involving the ~~Von~~ Mangoldt function:

The logarithm of the {{w|Riemann zeta function}} function can be expressed in terms of a {{w|Dirichlet series}} involving the von Mangoldt function:

<math>\displaystyle

@@ Line 71: / Line 71: @@
 Seeing that we notate the power as ''s'', it might become apparent where the Riemann zeta function will eventually show up.
-If ''s'' is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting cancels this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of {{sfrac|1|''n''}} for each prime power ''p''<sup>''n''</sup>. A somewhat peculiar but useful way to write the result of doing this is in terms of the {{w|Von Mangoldt function}}, an {{w|arithmetic function}} on positive integers which is equal to ln(''p'') on prime powers ''p''<sup>''n''</sup>, and is zero elsewhere. This is written using a capital lambda, as Λ(''n''), and in terms of it we can include prime powers in our error function as
+If ''s'' is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting cancels this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of {{sfrac|1|''n''}} for each prime power ''p''<sup>''n''</sup>. A somewhat peculiar but useful way to write the result of doing this is in terms of the {{w|von Mangoldt function}}, an {{w|arithmetic function}} on positive integers which is equal to ln(''p'') on prime powers ''p''<sup>''n''</sup>, and is zero elsewhere. This is written using a capital lambda, as Λ(''n''), and in terms of it we can include prime powers in our error function as
 <math>\displaystyle \xi_\infty(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\lfloor x \log_2 n \rceil^2}{n^s}</math>
@@ Line 89: / Line 89: @@
 This new function has the property that {{nowrap|F<sub>s</sub>(''x'') {{=}} F<sub>s</sub>(0) − E<sub>s</sub>(''x'')}}, so that all we have done is flip the sign of E<sub>s</sub>(''x'') and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum.
 Of more interest is the fact that it is a known mathematical function.
-The logarithm of the {{w|Riemann zeta function}} function can be expressed in terms of a {{w|Dirichlet series}} involving the Von Mangoldt function:
+The logarithm of the {{w|Riemann zeta function}} function can be expressed in terms of a {{w|Dirichlet series}} involving the von Mangoldt function:
 <math>\displaystyle