Summing It Up by Ash Avner; Gross Robert;

Author:Ash, Avner; Gross, Robert;
Language: eng
Format: epub, pdf
Publisher: Princeton University Press
Published: 2016-06-30T16:00:00+00:00

There is an ambiguity in the logarithm of a complex function. Remember that ez = ez+2πik for any integer k. If w = ez, and conditions are favorable, we can choose for log w one of the values z, z + 2πi, z – 2πi, and so on consistently as z varies, so long as z does not vary too much. This is called “choosing a branch” of the logarithm. We can and do choose a branch of log(ζ(s)) such that (10.8) is valid.

Next, we take the derivatives of both sides with respect to s. (Again, ignore convergence issues—the details work out.) Remember that the derivative of log(s) is 1/s, and use the chain rule. So, for any analytic function f(s) for which we can define a branch of the logarithm log f(s), we have . We also need to work out using the chain rule and the fact that the exponential function is its own derivative.

On the left-hand side, we get

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