# §6.16 Mathematical Applications

## §6.16(i) The Gibbs Phenomenon

Consider the Fourier series

6.16.1

The th partial sum is given by

where

By integration by parts

uniformly for . Hence, if is fixed and , then , 0, or according as , , or ; compare (6.2.14).

These limits are not approached uniformly, however. The first maximum of for positive occurs at and equals ; compare Figure 6.3.2. Hence if and , then the limiting value of overshoots by approximately 18%. Similarly if , then the limiting value of undershoots by approximately 10%, and so on. Compare Figure 6.16.1.

This nonuniformity of convergence is an illustration of the Gibbs phenomenon. It occurs with Fourier-series expansions of all piecewise continuous functions. See Carslaw (1930) for additional graphs and information.

Figure 6.16.1: Graph of , , , illustrating the Gibbs phenomenon.

## §6.16(ii) Number-Theoretic Significance of

If we assume Riemann’s hypothesis that all nonreal zeros of have real part of 25.10(i)), then

where is the number of primes less than or equal to . Compare §27.12 and Figure 6.16.2. See also Bays and Hudson (2000).

Figure 6.16.2: The logarithmic integral , together with vertical bars indicating the value of for .
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