Let $q$ be an odd integer, let $\tau$ be the order of $2$ modulo $q$ and let $\xi$ be a primitive $q$th root of unity. Upper bounds for $\sum_{k=1}^\tau \xi^{2^k}$ are proved in terms of the parameters $\mu$ and $\nu$ when $q$ diverges along sequences $S_{\mu,\nu}$ for which the quotient $\tau/\log_2 q$ belongs to the interval $[\mu,\nu]$, with $1\leq \mu$ and $\nu$ close enough to $1$.
Cancellation in a short exponential sum / G. Molteni. - In: JOURNAL OF NUMBER THEORY. - ISSN 0022-314X. - 130:9(2010 Sep), pp. 2011-2027. [10.1016/j.jnt.2010.03.002]
Cancellation in a short exponential sum
G. Molteni
2010
Abstract
Let $q$ be an odd integer, let $\tau$ be the order of $2$ modulo $q$ and let $\xi$ be a primitive $q$th root of unity. Upper bounds for $\sum_{k=1}^\tau \xi^{2^k}$ are proved in terms of the parameters $\mu$ and $\nu$ when $q$ diverges along sequences $S_{\mu,\nu}$ for which the quotient $\tau/\log_2 q$ belongs to the interval $[\mu,\nu]$, with $1\leq \mu$ and $\nu$ close enough to $1$.
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