Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line

Abstract Let 𝐹 be a linear combination (of a certain form) of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> N\geq 1 Dirichlet 𝐿-functions attached to even (or odd) primitive characters. Selberg proved that a positive proportion of non-trivial zeros of 𝐹 lie on the critical line. Our work here is to provide an explicit lower bound for this proportion. In particular, we show that the lower bound <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mn>2.16</m:mn> <m:mo lspace="0.222em" rspace="0.222em">×</m:mo> <m:msup> <m:mn>10</m:mn> <m:mrow> <m:mo>−</m:mo> <m:mn>6</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo>/</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>N</m:mi> <m:mo lspace="0.167em">⁢</m:mo> <m:mrow> <m:mi>log</m:mi> <m:mo lspace="0.167em">⁡</m:mo> <m:mi>N</m:mi> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> 2.16\times 10^{-6}/(N\log N) is admissible for large 𝑁.