The Riemann zeta function and tuning: Difference between revisions
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- \arctan\left(\frac{2t}{4n+1}\right)\right)</math> | - \arctan\left(\frac{2t}{4n+1}\right)\right)</math> | ||
Since the arctangent function is holomorphic in the strip with imaginary part between -1 and 1, it follows from the above formula, or arguing from the previous one, that θ is holomorphic in the strip with imaginary part between -1/2 and 1/2. It may be described for real arguments as an odd real analytic function of ''x'', increasing when |''x'' | Since the arctangent function is holomorphic in the strip with imaginary part between -1 and 1, it follows from the above formula, or arguing from the previous one, that θ is holomorphic in the strip with imaginary part between -1/2 and 1/2. It may be described for real arguments as an odd real analytic function of ''x'', increasing when {{nowrap|{{!}}''x''{{!}} > 6.29}}. Plots of it may be studied by use of the Wolfram [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelTheta online function plotter]. | ||
Using the theta and zeta functions, we define the [[Wikipedia:Z function|Z function]] as | Using the theta and zeta functions, we define the [[Wikipedia:Z function|Z function]] as | ||
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where the product is over all primes p. The product converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying ζ(s) by the corresponding factors {{nowrap|(1 − p<sup>−s</sup>)}} for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, {{nowrap|(1 − 2<sup>−s</sup>)ζ(s)}} is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3. | where the product is over all primes p. The product converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying ζ(s) by the corresponding factors {{nowrap|(1 − p<sup>−s</sup>)}} for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, {{nowrap|(1 − 2<sup>−s</sup>)ζ(s)}} is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3. | ||
Along the critical line, {{nowrap|1 − p^(−1/2 − it) | Along the critical line, {{nowrap|{{!}}1 − p^(−1/2 − it){{!}}}} may be written | ||
<math>\displaystyle{ | <math>\displaystyle{ | ||