On the Balasubramanian-Ramachandra method close to Re(s)=1

We study the problem on how to get good lower estimates for the integral $\int_T^{T+H} |ζ(σ+it)| dt,$ when $H \ll 1$ is small and $σ$ is close to $1$, as well as related integrals for other Dirichlet series, by using ideas related to the Balasubramanian-Ramachandra method. We use kernel-functions constructed by the Paley-Wiener theorem as well as the kernel function of Ramachandra. We also notice that the Fourier transform of Ramachandra's Kernel-function is in fact a $K$-Bessel function. This simplifies some aspects of Balasubramanian-Ramachandra method since it allows use of the theory of Bessel-functions.

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