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In this paper, we find all sums of two Fibonacci numbers which are close to a power of 2. As a corollary, we also determine all Lucas numbers close to a power of 2. The main tools used in this work are lower bounds for linear forms in logarithms due to Matveev and Dujella-Pethö version of the Baker-Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of Chern and Cui.

Given a number field K K of degree n K n_K and with absolute discriminant d K d_K , we obtain an explicit bound for the number N K ( T ) N_K(T) of non-trivial zeros (counted with multiplicity), with height at most T T , of the Dedekind zeta function ζ K ( s ) \zeta _K(s) of K K . More precisely, we show that for T ≥ 1 T \geq 1 , | N K ( T ) − T π log ⁡ ( d K ( T 2 π e ) n K ) | ≤ 0.228 ( log ⁡ d K + n K log ⁡ T ) + 23.108 n K + 4.520 , \begin{equation*} \Big | N_K (T) - \frac {T}{\pi } \log \Big ( d_K \Big ( \frac {T}{2\pi e}\Big )^{n_K}\Big )\Big | \le 0.228 (\log d_K + n_K \log T) + 23.108 n_K + 4.520, \end{equation*} which improves previous results of Kadiri and Ng, and Trudgian. The improvement is based on ideas from the recent work of Bennett et al. on counting zeros of Dirichlet L L -functions.

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Given a number field K of degree nK and with absolute discriminant dK , we obtain an explicit bound for the number NK (T ) of non-trivial zeros, with height at most T , of the Dedekind zeta function ζK (s) of K. More precisely, we show that NK (T ) − T π log dK T 2πe n K ≤ 0.228(log dK + nK log T ) + 23.108nK + 4.520, which improves previous results of Kadiri-Ng and Trudgian. The improvement is based on ideas from the recent work of Bennett et al. on counting zeros of Dirichlet L-functions.