6 Exponential, Logarithmic, Sine, and Cosine IntegralsApplications6.15 Sums6.17 Physical Applications

- §6.16(i) The Gibbs Phenomenon
- §6.16(ii) Number-Theoretic Significance of $\mathrm{li}\left(x\right)$

Consider the Fourier series

6.16.1 | $$ | ||

The $n$th partial sum is given by

6.16.2 | $${S}_{n}(x)=\sum _{k=0}^{n-1}\frac{\mathrm{sin}\left((2k+1)x\right)}{2k+1}=\frac{1}{2}{\int}_{0}^{x}\frac{\mathrm{sin}\left(2nt\right)}{\mathrm{sin}t}dt=\frac{1}{2}\mathrm{Si}\left(2nx\right)+{R}_{n}(x),$$ | ||

where

6.16.3 | $${R}_{n}(x)=\frac{1}{2}{\int}_{0}^{x}\left(\frac{1}{\mathrm{sin}t}-\frac{1}{t}\right)\mathrm{sin}\left(2nt\right)dt.$$ | ||

By integration by parts

6.16.4 | $${R}_{n}(x)=O\left({n}^{-1}\right),$$ | ||

$n\to \mathrm{\infty}$, | |||

uniformly for $x\in [-\pi ,\pi ]$. Hence, if $x$ is fixed and $n\to \mathrm{\infty}$, then ${S}_{n}(x)\to \frac{1}{4}\pi $, $0$, or $-\frac{1}{4}\pi $ according as $$, $x=0$, or $$; compare (6.2.14).

These limits are not approached uniformly, however. The first maximum of $\frac{1}{2}\mathrm{Si}\left(x\right)$ for positive $x$ occurs at $x=\pi $ and equals $(1.1789\mathrm{\dots})\times \frac{1}{4}\pi $; compare Figure 6.3.2. Hence if $x=\pi /(2n)$ and $n\to \mathrm{\infty}$, then the limiting value of ${S}_{n}(x)$ overshoots $\frac{1}{4}\pi $ by approximately 18%. Similarly if $x=\pi /n$, then the limiting value of ${S}_{n}(x)$ undershoots $\frac{1}{4}\pi $ 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.

If we assume Riemann’s hypothesis that all nonreal zeros of $\zeta \left(s\right)$ have real part of $\frac{1}{2}$ (§25.10(i)), then

6.16.5 | $$\mathrm{li}\left(x\right)-\pi (x)=O\left(\sqrt{x}\mathrm{ln}x\right),$$ | ||

$x\to \mathrm{\infty}$, | |||

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