Riv. Mat. Univ. Parma, Vol. 11, No. 2, 2020

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

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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.

References
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J. Kaczorowski, Axiomatic theory of L-functions: the Selberg class, in "Analytic Number Theory", C.I.M.E. Summer School, Cetraro (Italy) 2002, A. Perelli & C. Viola, eds., Lecture Notes in Math., 1891, Springer, Berlin, 2006, 133-209. MR2277660
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J. Kaczorowski and A. Perelli, On the structure of the Selberg class, II: Invariants and conjectures, J. Reine Angew. Math. 524 (2000), 73-96. MR1770604
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