= Euler product positivity for L-function nonvanishing
{c}
{title2=$\zeta(\sigma)^3|L(\sigma+it,\chi)|^4|L(\sigma+2it,\chi^2)|\ge1$}
For $\sigma>1$, logarithmic expansion of the <Euler product> reduces positivity to $3+4\cos\theta+\cos2\theta=2(1+\cos\theta)^2$. At <primes> dividing the modulus the zeta contribution is positive by itself. If the last factor is bounded at the line-one point, a zero of the middle factor would outweigh the zeta <pole> and force this product to zero. This proves the relevant <nonvanishing of Dirichlet L-functions on the line one>.
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