The sinc\operatorname{sinc} function is defined by

sinc(x)=sin(x)x. \operatorname{sinc}(x) = \frac{\sin (x)}{x}.

A standard contour integral tells us

sinc(x)  dx=π. \int _{-\infty }^\infty \operatorname{sinc}(x) \; \mathrm{d}x = \pi .

Alternatively, we can observe that

sinc(x)=1211eikt  dk. \operatorname{sinc}(x) = \frac{1}{2} \int _{-1}^1 e^{i k t}\; \mathrm{d}k.

So up to some factors sincx\operatorname{sinc}x is the Fourier transform of the indicator function of [1,1][-1, 1]. The preceding integral of sincx\operatorname{sinc}x can be thought of as the value of the Fourier transform at 00. Applying the Fourier inversion formula and carefully keeping track of the coefficients gives us the previous calculation.

David and Jonathan Borwein 1 observed that we also have

sinc(x)sinc(x3)  dx=π,sinc(x)sinc(x3)sinc(x5)  dx=π,sinc(x)sinc(x3)sinc(x5)sinc(x7)  dx=π. \begin{aligned} \int _{-\infty }^\infty \operatorname{sinc}(x) \operatorname{sinc}\left(\frac{x}{3}\right) \; \mathrm{d}x & = \pi ,\\ \int _{-\infty }^\infty \operatorname{sinc}(x) \operatorname{sinc}\left(\frac{x}{3}\right)\operatorname{sinc}\left(\frac{x}{5}\right) \; \mathrm{d}x & = \pi ,\\ \int _{-\infty }^\infty \operatorname{sinc}(x) \operatorname{sinc}\left(\frac{x}{3}\right) \operatorname{sinc}\left(\frac{x}{5}\right) \operatorname{sinc}\left(\frac{x}{7}\right) \; \mathrm{d}x & = \pi . \end{aligned}

This pattern holds up until

sinc(x)sinc(x3)sinc(x13)  dx=π. \int _{-\infty }^\infty \operatorname{sinc}(x) \operatorname{sinc}\left(\frac{x}{3}\right) \cdots \operatorname{sinc}\left(\frac{x}{13}\right) \; \mathrm{d}x = \pi .

Afterwards, we have

sinc(x)sinc(x3)sinc(x15)  dx=467807924713440738696537864469467807924720320453655260875000π. \int _{-\infty }^\infty \operatorname{sinc}(x) \operatorname{sinc}\left(\frac{x}{3}\right) \cdots \operatorname{sinc}\left(\frac{x}{15}\right) \; \mathrm{d}x = \frac{467807924713440738696537864469}{467807924720320453655260875000} \pi .

As we keep going on, the value continues to decrease.