A computational approach to the generalized Riemann hypothesis

In this paper, we investigate analytically and numerically the behavior of the Li coefficients $\lambda _{\chi } (n)$ of Dirichlet $L$-functions $L(s, \chi ) $ for large sets of positive integers $n$ and for different primitive characters $\chi $. Actually, we show that the generalized Riemann hypothesis for $L(s,\chi )$ is equivalent to a certain expression of the Li coefficients in terms of a sum of Chebyshev polynomials. The numerical computations based on two different representations of the Li coefficients in terms of a sum over primes or a certain sum of Chebyshev polynomials suggest that the Li coefficients are positive and increasing for $n\geq 1$.