A note on large values of Dirichlet L -functions for characters of fixed order at 1 divided by 2 less than sigma less than or slanted equals 1 1 / 2 < σ ⩽ 1 $1/2\lt\sigma\leqslant 1$

Source abstract

Abstract In this paper, we use a simple argument to show the existence of large values of conjecturally sharp size for Dirichlet L -functions attached to primitive characters of fixed order at sigma element of left parenthesis 1 divided by 2 comma 1 right bracket σ ∈ ( 1 / 2 , 1 ] $\sigma\in (1/2, 1]$ . More precisely, for every fixed integer g greater than or slanted equals 2 g ⩾ 2 $g\geqslant 2$ we prove the existence of a primitive character chi χ $\chi$ of order g and conductor upper Q equivalent to x Q ≍ x $Q\asymp x$ such that StartAbsoluteValue upper L left parenthesis 1 comma chi right parenthesis EndAbsoluteValue greater than or slanted equals e Superscript gamma Baseline left parenthesis log log x plus log log log x minus log left parenthesis 2 log g right parenthesis plus o left parenthesis 1 right parenthesis right parenthesis period | L ( 1 , χ ) | ⩾ e γ ( log ⁡ log ⁡ x + log ⁡ log ⁡ log ⁡ x − log ⁡ ( 2 log ⁡ g ) + o ( 1 ) ) . $|L(1,\chi)|\geqslant e^\gamma\left(\log\log x+\log\log\log x-\log(2\log g)+o(1)\right)\!.$ We also show that for every fixed 1 divided by 2 less than sigma less than 1 1 / 2 < σ < 1 $1/2\lt\sigma\lt1$ there exists a primitive character chi χ $\chi$ of order g and conductor upper Q equivalent to x Q ≍ x $Q\asymp x$ such that log StartAbsoluteValue upper L left parenthesis sigma comma chi right parenthesis EndAbsoluteValue greater than or slanted equals left parenthesis upper C Subscript g Baseline left parenthesis sigma right parenthesis plus o left parenthesis 1 right parenthesis right parenthesis left parenthesis log x right parenthesis Superscript 1 minus sigma Baseline left parenthesis log log x right parenthesis Superscript negative sigma Baseline comma log ⁡ | L ( σ , χ ) | ⩾ ( C g ( σ ) + o ( 1 ) ) ( log ⁡ x ) 1 − σ ( log ⁡ log ⁡ x ) − σ , $\log |L(\sigma,\chi)|\geqslant\left(C_g(\sigma)+o(1)\right)(\!\log x)^{1-\sigma}(\!\log\log x)^{-\sigma},$ for some explicit positive constant upper C Subscript g Baseline left parenthesis sigma right parenthesis period C g ( σ ) . $C_g(\sigma).$ Previously, such bounds were known only conditionally on the Generalised Riemann Hypothesis, and even then only in the special cases g equals 2 g = 2 $g=2$ and g equals 3 g = 3 $g=3$ .