An extension of the Lehmers’ picturesque exponential sums. (English) Zbl 0781.11031

Apostol, Tom M.

Math. Comput. 61, No. 203, 25-28 (1993).

Let \(S_ j(m)=\sum^ m_{n=0} \zeta_ k^{b(n)+jn}\), \(j\geq 1\), where \(\zeta_ k=\exp(2\pi i/k)\) and \(b(n)\) is the sum of the digits of \(n\) when written to the base \(k\), where \(k\) is a fixed integer \(\geq 2\). In [Am. Math. Mon. 86, 725-733 (1979; Zbl 0433.10024)] D. H. Lehmer and E. Lehmer proved the so-called ‘Rotation Theorem’, \[ S_ j(nk^ 2+ d_ 1k+ d_ 0)= \zeta_ k^{b(n)} S_ j(d_ 1 k+d_ 0), \qquad n=d_ 0+d_ 1k+\cdots\;. \] In the present paper the author replaces the factor \(\zeta_ k^ j\) in \(S_ j(m)\) by an arbitrary complex number. For this more general exponential sum he proves a similar result which he calls ‘Rotation Plus Translation Theorem’.

Reviewer: J.Hinz (Marburg)

MSC:

11L03

Trigonometric and exponential sums (general theory)

Keywords:

exponential sum

Citations:

Zbl 0433.10024