Page 40

chalkdust In the form above, the representation is not very useful. Far beer is a truncation of the sum at some carefully chosen n = N(x), together with a big-O error for the remainder of the terms. There is an art to deciding the truncation point so as to reveal dominant behaviour of the expression whilst keeping the error in check.

Closing in on α Where does the problem stand? Thanks to Hardy we have a lower bound α > 14 . In fact, from computational evidence and heuristic arguments this is expected to be exactly on the money—we believe that α = 41 . If true, the first person to show an upper bound that is also 14 will have solved Dirichlet’s divisor problem. However, 150 years of dedicated work on the upper bound can be read in the decelerating sequence 33 27 15 12 346 35 7 , 82 , 46 , 37 , 1067 , 108 , 22 . The current best has stood since 2003, when a of improvements: 12 , 13 , 100 131 ≈ 0.3149. contribution by Martin Huxley showed that α 6 416

The upper bound (blue) on α, slowly geing closer to

1 4


We have confined α to the interval [0.25, 0.3149]. At the current rate of progress, we are still a long time from squeezing this interval to one number, thus pinning down the precise value of α. Well over a century on, Dirichlet’s challenge is still to be met.

James Cann is a PhD student at UCL and is aached to the London School of Geometry & Number Theory. Reach him by post at ‘UCL Department of Mathematics’, or @jame5cann

My favourite shape

Sierpinski triangle Nikoleta Kalaydzhieva

My favourite shape is the Sierpinski triangle. It is one of the most basic fractal shapes, but appears in various mathematical areas. What I find fascinating about it is how many different ways there are for constructing it. For example, you could use a methodical geometric approach by inscribing a similar triangle in the original one via its midpoints and iterating. Another, more intriguing construction, is via the Chaos game. You can even construct it using basic algebra, by shading the odd numbers in Pascal’s triangle.


Chalkdust, Issue 03  

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