On the value distribution of two Dirichlet $\boldsymbol L$-functions

Abstract

We look at the values of two Dirichlet L-functions at the Riemann zeros (or a horizontal shift of them).Off the critical line we show that for a positive proportion of these points the pairs of values of the two L-functions are linearly independent over R, which, in particular, means that their arguments are different. On the critical line we show that, up to height T , the values are different for cT of the Riemann zeros for some positive c.the distribution of their zeros. In 1976 Fujii [3] showed that a positive proportion of zeros of L(s, ψ)L(s, χ) are distinct, where the characters are primitive and not necessarily of distinct moduli. A zero of the product is said to be distinct if it is a zero of only one of the two, or if it is a zero of both then it occurs with different multiplicities for each function. It is, in fact, believed that all zeros of Dirichlet L-functions to primitive characters are simple, and that two L-functions with distinct primitive characters do not share any non-trivial zeros at all. This comes from the Grand Simplicity Hypothesis (GSH), see [18]. The hypothesis is that the set {γ | L( 1 2 + iγ, χ) = 0 and χ is primitive} is linearly independent over Q. Since we are counting with multiplicities, it is implicit in the statement of the GSH that all zeros of Dirichlet L-functions are simple, and that γ = 0, i.e. L( 1 2 , χ) = 0. A similar result is expected for an even bigger class of functions. R. Murty and K. Murty [16] proved that two functions of the Selberg class S cannot share too many zeros (counted with multiplicity). They show that if F , G ∈ S then F = G provided thatwhere Z F (T ) denotes the set of zeros of F (s) in the region Re s ≥ 1/2 and |Im s| ≤ T , and ∆ is the symmetric difference. In 1986 Conrey et al. [1] proved that the Dedekind zeta function of a quadratic number field has infinitely many simple zeros. They were, however, unable to obtain the result for a positive proportion of the zeros.Apart from looking at the zeros, there has also been investigation into the a-values of ζ and L(s, χ), that is, the distribution of s such that ζ(s) = a (or L(s, χ) = a) for some fixed a ∈ C. In [6] Garunkštis and Steuding prove a discrete average for ζ ′ over the a-values of ζ, which implies that there are infinitely many simple a-points in the critical strip. On the critical line, however, we do not even know whether there are infinitely many a-points. For further results on

Discrete mean square of the Riemann zeta-function over imaginary parts of its zeros

“…Laaksonen and Petridis [19] investigated a similar sum to the sum of Theorems From this Laaksonen and Petridis [19] derived the result that, under RH, for a positive proportion of non-trivial zeros of ζ(s) with γ > 0, the values of the Dirichlet L-functions L(σ +iγ, χ 1 ) and L(σ +iγ, χ 2 ) are linearly independent over R.…”