On the Backlund equivalent for the Lindelöf hypothesis

Abstract

Backlund showed that the Lindelöf hypothesis (LH) for the Riemann zeta-function is equivalent to some regularity of the distribution of zeros to the right from the critical line. We generalize the Backlund equivalent, by showing that LH is equivalent to the same type regularity of the distribution of any fixed complex value (not only zero). This generalized Backlund equivalent also can be applied for the Lerch zeta-function and, in our opinion, supports the idea that the Lindelöf hypothesis also is reasonable for zeta functions without the Euler product (usually having zeros off the critical line). Further we show that this generalized Backlund equivalent for LH can be formulated for zeta functions of the Selberg class and for the Selberg zeta-function, for which the Riemann hypothesis is true.