An alternative to Riemann-Siegel type formulas (Q2792353)

In this paper, simple unsmoothed formulas to compute the Riemann zeta function \(\zeta(s)\), and Dirichlet \(L\)-functions to a powerful modulus, are obtained by using Taylor expansions and geometric series. Let \(q(s) = |s| + 3\) be the analytic conductor of the zeta function. We choose integers \(u_0 \geq 1\), \(v_0 \geq u_0\), and \(M \geq v_0\), and construct sequences \(K_r = \lceil v_r/u_0\rceil\) and \(v_{r+1} = v_r + K_r\) for \(0 \leq r < R\), where \(R = R(v_0, u_0, M)\) is the largest integer such that \(v_R < M\). We define \(K_R = \min\{\lceil v_R/u_0 \rceil , M-vR\}\), so that \(v_R+1 = M\). We consider the functions NEWLINE\[NEWLINEf_s(z) = \frac{e^{sz}}{(1+z)^s}, \;\;\;g_K(s) = \frac{e^{Kz}-1}{e^z-1}, \;z\not \in 2\pi \mathbb ZNEWLINE\]NEWLINE and we let NEWLINE\[NEWLINEBr(s, m) = \sum_{j=0}^m \frac{f_s^{(j)}(0)}{j!} \frac {g^{(j)}_{K_r}(-s/vr)}{v_r^j}NEWLINE\]NEWLINE and NEWLINE\[NEWLINEB_M(s, u_0, v_0) = \sum_{r=0}^R v_r^{-\sigma} \min\{g_{K_r}(-\sigma/v_r), |\csc(t/(2vr))|\}.NEWLINE\]NEWLINE Further, we put NEWLINE\[NEWLINE\varepsilon(s,u) = \frac{3.5 e^{0.78(m+1)}|s|^{(m+1)/2}}{(m+1)^{(m+1)/2}}NEWLINE\]NEWLINE for \( m \leq |s|/4\) and NEWLINE\[NEWLINE\varepsilon(s,u) =\frac{2^m e^{0,194|s|}}{u^m}NEWLINE\]NEWLINE for \(m > |s|/4\).NEWLINENEWLINEThe main result of the paper is the following: Given \(s = \sigma+i t\) with \(\sigma > 0\), let \(u_0\) and \(v_0\) be any integers satisfying \(v_0 \geq u_0 \geq 2 \max\{6, \sqrt{q(s)}, \sigma\}\). Then, for any integers \(M \geq v_0\) and \(m \geq 0\) we have: NEWLINE\[NEWLINE\zeta(s) = \sum_{n=1}^{v_0-1}\frac{1}{n^s}+\sum_{r=0}^R \frac{B_r(s,m)}{v_r^s}+\frac{M^{-s}}{2}+ \frac{M^{1-s}}{s-1}+T_{M,m}(s, u_0, v_0) + R_M(s),NEWLINE\]NEWLINE where \(|T_{M,m}(s, u_0, v_0)| \leq \varepsilon_m(s, u_0) B_M(s, u_0, v_0)\) and \(R < 2u_0 \log(M/v0) + 1\).NEWLINENEWLINEThe method of the proof of this result is adapted in the case of Dirichlet \(L\)-functions and a formula for the function \(L(s,\chi)\) is obtained.