On the argument of $L$-functions

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 $S(t,\pi )=\frac{1}{\pi }argL(\frac{1}{2}+it,\pi )$, which is the derivative of $S_1(\cdot ,\pi )$ at $t$.

12 pages

References in corpus (3)

• Bounding $\zeta (s)$ in the critical strip
• Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function
• A note on the zeros of zeta and $L$-functions

Cited by in corpus (3)

• A note on the zeros of zeta and $L$-functions
• Bandlimited approximations and estimates for the Riemann zeta-function
• Bounding $S_n(t)$ on the Riemann hypothesis