On the Li coefficients for the Dirichlet \(L\)-functions
The paper is devoted to the investigation of the Li coefficients for the Dirichlet \(L\)-functions \(\lambda_\chi(n)\) and. As usual, denote by \(\zeta(s)\) the Riemann zeta-function. Let \(\Lambda(k)\) be the von Mangoldt function, and \(\chi\) be a primitive Dirichlet character \(\mod q\), \(q>1\). Then it is proved that, for the numbers \(\lambda_\chi(n)\), the following arithmetic formula \[ \lambda_\chi(n)=-\sum_{j=1}^{n}\binom{n}{j} \frac{(-1)^{j-1}}{(j-1)!}\sum_{k=1}^{+\infty}\frac{\Lambda(k)}{k}\chi(k)(\log k)^{j-1} +\frac{n}{2}\big(\log \frac{q}{\pi}-\gamma\big)+\tau_\chi(n) \] holds. Here, for \(\chi(-1)=1\), \(\tau_\chi(n)=\sum_{j=2}^{n}\binom{n}{j}(-1)^j\big(1-\frac{1}{2^j}\big)\zeta(j)-\frac{n}{2}\sum_{l=1}^{+\infty}\frac{1}{l(2l-1)}\), and, for \(\chi(-1)=-1\), \(\tau_\chi(n)=\sum_{j=2}^{n}\binom{j}{n}(-1)^{j}2^{-j}\zeta(j)\). Some similar arithmetic formulas are given by \textit{M. Coffey} [Math. Phys. Anal. Geom. 8, No. 3, 211--255 (2005; Zbl 1097.11042)] and \textit{X.-J. Li} [J. Math. 48, No. 2, 491--503 (2004; Zbl 1061.11048)]. The main result of the paper says that the Riemann hypothesis is equivalent to asymptotic formulas for the Li coefficients \(\lambda_\chi(n)\), i.e., if \(\lambda_\chi(n)=\frac{1}{2}n\log n+\big(\frac{1}{2}(\gamma-1)+\frac{1}{2}\log(\frac{q}{\pi})\big)n +O(\sqrt{n}\log n)\) or \(\lambda_\chi(n)=\frac{1}{2}n \log n+\big(\frac{1}{2}(\gamma-1)+\frac{1}{2}\log(\frac{q}{\pi})\big)n+n y_n\), where \((y_n)\in l_2\) and \(\gamma\) is the Euler constant. Also, it is proved that the Riemann hypothesis for the Dirichlet \(L\)-functions holds if and only if the coefficients \(\lambda_\chi(n)\) are increasing in \(n\), and, using the Bell polynomials, unconditionally it is proved that the first Li coefficients are increasing. As an application, a probabilistic interpretation of Li coefficients is given.
scientific article; zbMATH DE number 6176420
scientific article