Character sums and the Riemann hypothesis (Q6595590)

Let \N\[\Nf(x)=\sum_{n=1}^\infty \frac{\lambda(n) \sin (2\pi nx)}{n^2}, \N\]\Nwhere \(\lambda(n)\) is the Liouville lambda-function which is a completely multiplicative function and takes the value \(-1\) on primes so that \( \lambda(p_1^{e_1}\ldots p_r^{e_r})=(-1)^{e_1+\cdots+e_r}\).\N\NIn this paper under review, it is first shown that if \(f(x)\geq 0\) for \(0\leq x\leq 1/4\) then the Riemann hypothesis (RH) is true. Furthermore, The bound \(1/4\) in the above theorem can be replaced by any positive constant. So the real issue is trying to prove that \(f(x) > 0 \) for small positive \(x\). Since \(\lambda\) is a completely multiplicative function and takes the values \(\pm 1 \), the author considers a similar sum with quadratic Dirichlet character defined by \N\[\Nf_q(x)=\sum_{n=1}^\infty \frac{\left(\frac{n}{q}\right)\sin (2\pi nx)}{n^2}, \N\]\Nwhere \(q\) is a prime number and \(\left(\frac{n}{q}\right)\) is the Legendre symbol. Then it is shown that if \(f_q(x)\geq 0\) for all \(0\leq x\leq 1/4\) and all primes \(q\) congruent to \(3\) modulo \(8\) then the Riemann hypothesis is true. Finally, with the use of numerical values of \(f_{163}(x)\), it is shown that \(f(x)\geq 0\) for \(0.043\leq x\leq 0.25\).