Character sums and the series \(L(1,\chi)\) (Q2747269)

For \(k>1\), let \(\chi \) be a real primitive character \(\pmod{k}\), and let \(T(v,j,\chi) =\sum_{n=j+1}^{j+k}\chi (n)(vk+n)^{-1}\) be a segment of length \(k\) of the series \(L(1,\chi)\). The author compares the signs of \(T(v,j,\chi)\) and certain other sums. The result for even characters is that if \(M =\sum_{m=1}^j \chi (m) \neq 0\) and \(v \geq \max (1, \sqrt{k}/(4|M|)\), then \(T(v,j,\chi)\) and \(M\) are of opposite sign. In the case of odd characters, let \(h(-k)\) be the class number of the field \(\mathbb Q(\sqrt{-k})\) and define now \(M=h(-k)-\sum_{m=1}^j \chi (m)\). Then \(T(v,j,\chi)\) and \(M\) are of the same sign with the same assumptions as above.