Karol Gierszewski [a]
Some remarks on twists and Euler products in the Selberg class
Pages: 293-299
Received: 28 November 2019
Accepted in revised form: 4 May 2020
Mathematics Subject Classification (2010): 11M41, 11F66.
Keywords: Selberg class, Converse theorem, Twists, Euler products.
Author address:
[a]: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
Abstract: We present a new variant of the method of J. Kaczorowski and A. Perelli for obtaining the converse theorem via Euler products in the Selberg class. A standard method requires proving a weak zero-density estimate for functions from the Selberg class in the whole half-plane \(\sigma > 1/2 \). Such an estimate is not known in general, even for the classical \(L\)-functions of high degrees, to hold. Our modification, while providing the same result, does not need as an ingredient a weak zero-density estimates for functions in the Selberg class.