chalkdust

Exploiting symmetry This was not good enough for Dirichlet. He wanted to pin down the constant—that number floating around in the O(1) envelope above. With some rather clever counting he exploited a natural symmetry of the hyperbola XY = x so as to make fewer than the x lots of O(1) approximations we made in (3). If we swap the X and Y axes the hyperbola is le unchanged: there is a reflective symmetry about the line X = Y. Counting the laice points lying below the hyperbola but vertically above the X-axis coordinates √ 1, 2, . . . , [ x] is exactly the same as counting the laice points lying below the hyperbola but horizontally to the right of the Y-axis coordi√ nates 1, 2, . . . , [ x]. Combined, these counts cover all of the laice points under the hyperbola. Except that we have over-counted the √ √ [ x] × [ x] laice points lying in the square below the hyperbola which is sent to itself under the symmetry. In symbols: ∑ [ x ] [√ ]2 D(x) = 2 − x . √ n

x

By symmetry, the pink and cyan areas contain the same number of points.

n6 x

Arithmetic along the lines of (3) transforms this into: √ D(x) = x log x + (2γ − 1)x + O( x). √ Dirichlet had pinned down the constant. Upon dividing both sides above by x, the O( x) turns into O( √1x ), which vanishes as x becomes large. We have shown that the mean value of the divisor function d(n) taken over n 6 x, for large x, is log x + 2γ − 1.

Dirichlet’s divisor problem Well, I feel content. Don’t you? But not Dirichlet. Never Dirichlet. Despite a decent amount of work, a rather sneaky argument and answering all of the questions that he had asked, he wanted √ to further dissect the O( x). He was convinced that he was being far more imprecise than he needed to be: like approximating [x 2 ] by x 2 + O(x), when in fact we can be sure that it is x 2 + O(1). Writing Δ(x) = D(x) − x log x − (2γ − 1)x, our eﬀorts so far have shown that Δ(x) = O(x1/2 ). In his leer to Kronecker, Dirichlet hinted that he could replace O(x1/2 ) by something significantly more precise. Nothing in all of Dirichlet’s chalkdustmagazine.com

36

Chalkdust, Issue 03

Popular mathematics magazine from UCL