Talk:Selberg class

Comments

I'm confused about the need for an ε in condition ii). Should this not be "$a_{n}=O(n^{r+\epsilon })$ for some fixed positive real number r and any ε>0"?

Is (iii) right? It looks backwards to me.

I'm used to seeing phi(z) = gamma factor times original Dirichlet series, then the functional equation is phi(z) = <some number of absolute value 1> times the complex conjugate of phi(1 - complex conjugate of z) Virginia-American (talk) 02:48, 25 February 2008 (UTC)

The second reference (Conrey & Ghosh) state it the way I would expect. I will edit the article Virginia-American (talk) 03:21, 25 February 2008 (UTC)

The sentence "The condition that $\theta <1/2$ is important, as the $\theta =1/2$ case includes the Dirichlet eta-function, which violates the Riemann hypothesis." is not true: the Dirichlet eta does not violate RH, it just doesn't have a pole at $s=1$. The function $\zeta \left({\frac {s+1}{3}}\right)$ violates RH as noted in paper [2]. — Preceding unsigned comment added by 95.168.124.177 (talk) 12:25, 9 November 2018 (UTC)

Examples

The statement about F^χ can't be right if χ is nontrivial. Take for example $F(s)=L(s,{\overline {\chi }})$, which is in the Selberg class. Then $F^{\chi }(s)=L(s,{\overline {\chi }}\chi )$ is the Dirichlet L-function of an imprimitive principal character, and therefore not in the Selberg class.—Emil J. 23:41, 11 June 2014 (UTC)

I checked Selberg’s paper. The definition of F^χ is correct, however, he does not say that F^χ is S. He says that if F^χ is in S, then something happens (conjecturally).—Emil J. 16:23, 12 June 2014 (UTC)