The Riemann zeta function and tuning: Difference between revisions

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}</math>

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~~ {{nowrap|''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''−''s'')}} 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−''s'')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 one, while at {{nowrap|''s'' {{=}} 1}} 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''−''s'')}} 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−''s'')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:

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 }</math>
-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 {{nowrap|''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 one, while at {{nowrap|''s'' {{=}} 1}} 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: