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arXiv · 1504.01833

On the argument of $L$-functions

Emanuel Carneiro·Renan Finder
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Abstract

For $L(\cdot,\pi)$ in a large class of $L$-functions, assuming the generalized Riemann hypothesis, we show an explicit bound for the function $S_1(t,\pi)=\frac{1}{\pi}\int_{1/2}^\infty\log|L(\sigma+it,\pi)|\,d\sigma$, expressed in terms of its analytic conductor. This enables us to give an alternative proof of the most recent (conditional) bound for $S(t,\pi)=\frac{1}{\pi} \,arg\,L(\tfrac12+it,\pi)$, which is the derivative of $S_1(\cdot,\pi)$ at $t$.

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Emanuel Carneiro, Renan Finder. 2015-04-08. On the argument of $L$-functions. https://doi.org/10.1007/s00574-015-0105-y

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