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We prove that, for every sufficiently large modulus $q\not\equiv2\bmod4$, at least $3/8-o(1)$ of the primitive Dirichlet characters $\chi$ modulo $q$ have $L(1/2,\chi)\ne0$. We also establish stronger proportions for moduli $q$ satisfying certain arithmetic conditions.
Let $K_1,\cdots,K_r$ be fixed number fields, and let $L$ be the compositum of their Galois closures. Assuming GRH for $\zeta_L$, we prove a sharp lower bound for products of shifted Dedekind zeta functions on the critical line, for arbitrary fixed positive real exponents and uniformly for shifts of size at most $T/2$....
We prove that, for either a Steinhaus random multiplicative function or a Rademacher random multiplicative function $f$, and every $\varepsilon>0$, almost surely $|\sum_{n\le x}f(n)|\ll_{\varepsilon,f}\sqrt{x}(\log\log x)^{1/4+\varepsilon}$. Together with Harper's almost sure lower bound, this determines the sharp loga...
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