Riemann zeta function: Difference between revisions
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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 | Along the critical line, {{nowrap|{{!}}1 − p<sup>−1/2 − it</sup>{{!}}}} may be written | ||
<math>\displaystyle{ | <math>\displaystyle{ | ||