The Riemann zeta function and tuning: Difference between revisions

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where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.

Another consequence of the above definition which might be objected to is that it results in a function with a [[Wikipedia:~~Continuous_function~~#~~Relation_to_differentiability_and_integrability~~|discontinuous derivative]], whereas a smooth function be preferred. The function ⌊x⌉<sup>2</sup> is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is {{nowrap|1 − cos(2π''x'')}}, which is a smooth and in fact an ~~[[Wikipedia:entire function~~|entire function]]. Let us therefore now define for any {{nowrap|''s'' > 1}}:

Another consequence of the above definition which might be objected to is that it results in a function with a [[Wikipedia:Continuous function#Relation to differentiability and integrability|discontinuous derivative]], whereas a smooth function be preferred. The function ⌊''x''⌉<sup>2</sup> is quadratically increasing near integer values of ''x'', and is periodic with period 1. Another function with these same properties is {{nowrap|1 − cos(2π''x'')}}, which is a smooth and in fact an {{w|entire function}}. Let us therefore now define for any {{nowrap|''s'' > 1}}:

<math>\displaystyle E_s(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x \log_2 n)}{n^s}</math>

For any fixed {{nowrap|''s'' > 1}} this gives a real ~~[[Wikipedia:analytic function~~|analytic function]] defined for all ''x'', and hence with all the smoothness properties we could desire.

For any fixed {{nowrap|''s'' > 1}} this gives a real {{w|analytic function}} defined for all ''x'', and hence with all the smoothness properties we could desire.

We can clean up this definition to get essentially the same function:

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 where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.
-Another consequence of the above definition which might be objected to is that it results in a function with a [[Wikipedia:Continuous_function#Relation_to_differentiability_and_integrability|discontinuous derivative]], whereas a smooth function be preferred. The function &#8970;x&#8969;<sup>2</sup> is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is {{nowrap|1 &minus; cos(2&pi;''x'')}}, which is a smooth and in fact an [[Wikipedia:entire function|entire function]]. Let us therefore now define for any {{nowrap|''s'' &gt; 1}}:
+Another consequence of the above definition which might be objected to is that it results in a function with a [[Wikipedia:Continuous function#Relation to differentiability and integrability|discontinuous derivative]], whereas a smooth function be preferred. The function &#8970;''x''&#8969;<sup>2</sup> is quadratically increasing near integer values of ''x'', and is periodic with period 1. Another function with these same properties is {{nowrap|1 &minus; cos(2&pi;''x'')}}, which is a smooth and in fact an {{w|entire function}}. Let us therefore now define for any {{nowrap|''s'' &gt; 1}}:
 <math>\displaystyle E_s(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x \log_2 n)}{n^s}</math>
-For any fixed {{nowrap|''s'' &gt; 1}} this gives a real [[Wikipedia:analytic function|analytic function]] defined for all ''x'', and hence with all the smoothness properties we could desire.
+For any fixed {{nowrap|''s'' &gt; 1}} this gives a real {{w|analytic function}} defined for all ''x'', and hence with all the smoothness properties we could desire.
 We can clean up this definition to get essentially the same function: