Distribution of zeros of Dirichlet \(L\)-functions and an explicit formula for \(\psi(t,\chi)\) (Q2781464)

The present paper deals with numerical results on the zero-distribution of Dirichlet \(L\)-functions \(L(s,\chi)\) and related questions. Continuing some results of \textit{K. S. McCurley} [J. Number Theory 19, 7-32 (1984; Zbl 0536.10035)], the authors give explicit numerical bounds for the zeros of \(L(s,\chi)\), in particular for the possible Siegel zero, and for the zero-density of \(L(s,\chi)\) near the vertical line \(\Re s=1\); the proofs use ideas of \textit{S. N. Graham} [Acta Arith. 39, 163-179 (1981; Zbl 0464.10032)] and of \textit{D. R. Heath-Brown} [Proc. Lond. Math. Soc. (3) 64, 265-338 (1992; Zbl 0739.11033)]. These results yield an explicit formula with a numerical value for the constant in the error term: suppose that \(N\) is an integer \(\geq\exp(2000)\), and \(t\in[0.001N,N]\), then NEWLINE\[NEWLINE \psi(t,\chi):=\sum_{n\leq t}\Lambda(n)\chi(n) =\delta(\chi)t-\sum_{\beta\geq 1/2,|\gamma|\leq T} {t^\rho\over \rho}+1.3804\theta t(\log N)^{-13},NEWLINE\]NEWLINE where \(\theta\) is a complex number of absolute value \(\leq 1\), \(\Lambda(n)\) is the von Mangoldt-function, the sum is taken over all nontrivial zeros \(\rho=\beta+i\gamma\) of \(L(s,\chi)\), and \(\delta(\chi)=1\) if \(\chi\) is the principal character, respectively \(\delta(\chi)=0\) otherwise. NEWLINENEWLINENEWLINEFurther, the authors announce an interesting application in the three primes Goldbach conjecture, namely that every odd integer exceeding \(\exp(3100)\) is a sum of three primes.