By Mathematical Association of America
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Additional resources for The American Mathematical Monthly,Volume 119, Number 1, January 2012
Sample text
According to Bradley [1], Euler arrived at a complete understanding of the complex logarithmic function between 1743 and 1746. Apparently, at the time of Euler’s 1741 definitive evaluation of 1/k 2 , he still harbored some uncertainty about the meaning of ln(−1). This may be evidence in favor of Roy’s idea. If Euler was aware of (5) prior to 1746, his confidence in Lewin’s argument would likely have been undermined by uncertainties about the logarithms of negative quantities. If his discovery of (5) came later than 1746, his interest in Lewin’s argument might have been diluted by the feeling that 1/k 2 was well established.
In particular, if we define F(z) = 1 − ln(1 − z) z if z = 0, otherwise, then F is analytic in . Of course away from the origin, F is analytic throughout because it is the product of analytic functions. On the other hand, for |z| < 1 we have the series representation − ln(1 − z) = z + z2 z3 + + ··· . 2 3 This shows that − ln(1 − z) z z2 =1+ + + ··· z 2 3 (8) for z = 0. The function F agrees in a neighborhood of zero with the series on the right, and so is analytic there. Now we can define Li2 (z).
What Zariski discovered in his fundamental paper [12] is that the solution of the Riemann-Roch problem for D was strongly controlled by its positive part. More precisely, letting P denote the positive part, he showed that lim n→∞ dim |n D| = P · P. 2) To illustrate this formula, we present two examples. 1. Let D be an effective divisor of degree d in the plane. The linear system |n D| consists of all effective divisors of degree nd, and thus has dimension nd+2 − 1. 2 By B´ezout’s theorem, the intersection of D with any irreducible curve is positive; hence D is nef, and thus its positive part is D itself.