Riv. Mat. Univ. Parma, Vol. 12, No. 1, 2021
Jerzy Kaczorowski [a,b] and Alberto Perelli [c]
Analytic properties of the standard twist of \(L\)-functions
Pages: 125-141
Received: 24 November 2019
Accepted in revised form: 15 April 2020
Mathematics Subject Classification (2010): 11M41.
Keywords: \(L\)-functions, standard twist, convexity bounds,
distribution of zeros, nonlinear exponential sums, Selberg class.
Authors address:
[a]: Faculty of Mathematics and Computer Science, A. Mickiewicz University, 61-614 Poznań, Poland
[b]: Institute of Mathematics, Polish Academy of Sciences, 00-956 Warsaw, Poland
[c]: Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova, Italy
This research was partially supported by the Istituto Nazionale di
Alta Matematica, by the MIUR grant
PRIN-2015 ''Number Theory and Arithmetic Geometry''
and by grant 2017/25/B/ST1/00208 ''Analytic methods in number theory''
from the National Science Centre, Poland
Full Text (PDF)
Abstract:
We study some consequences of the functional equation satisfied by the
standard twist \(F(s,\alpha)\) of the \(L\)-functions \(F(s)\) from the
extended Selberg class. The shape of such a functional equation differs
significantly from the classical one of Riemann-type satisfied by
\(F(s)\); for example, it contains an error term which can identically
vanish only in some special but well described cases. In this paper
we show that this unusual functional equation can nevertheless be
used to investigate convexity bounds, asymptotic formulae for the
average of the coefficients and distribution of zeros of \(F(s,\alpha)\).
References
- [1]
-
J. B. Conrey and A. Ghosh,
On the Selberg class of Dirichlet series: small degrees,
Duke Math. J. 72 (1993), 673-693.
MR1253620
- [2]
-
M. N. Huxley,
Area, lattice points and exponential sums,
London Math. Soc. Monogr. (N.S.), 13,
Oxford University Press, New York, 1996.
MR1420620
- [3]
-
J. Kaczorowski,
Axiomatic theory of \(L\)-functions: the Selberg class,
in ''Analytic Number Theory'', C.I.M.E. Summer School, (Cetraro, Italy, 2002),
ed. by A. Perelli and C. Viola,
Lecture Notes in Math., 1891, Springer, Berlin, 2006, 133-209.
MR2277660
- [4]
-
J. Kaczorowski and A. Perelli,
On the structure of the Selberg class, I: \(0\leq d \leq 1\),
Acta Math. 182 (1999), 207-241.
MR1710182
- [5]
-
J. Kaczorowski and A. Perelli,
The Selberg class: a survey, in ''Number Theory in Progress'', 2,
Proc. Conf. in Honor of A. Schinzel, ed. by K. Györy et al.,
Walter de Gruyter & Co., Berlin, 1999, 953-992.
MR1689554
- [6]
-
J. Kaczorowski and A. Perelli,
On the structure of the Selberg class, VI: non-linear twists,
Acta Arith. 116 (2005), 315-341.
MR2110507
- [7]
-
J. Kaczorowski and A. Perelli,
On the structure of the Selberg class, VII: \( 1 < d < 2 \),
Ann. of Math. 173 (2011), 1397-1441.
MR2800717
- [8]
-
J. Kaczorowski and A. Perelli,
Twists and resonance of \(L\)-functions, I,
J. Eur. Math. Soc. (JEMS) 18 (2016), 1349-1389.
MR3500839
- [9]
-
J. Kaczorowski and A. Perelli,
On the standard twist of the \(L\)-functions of half-integral weight cusp forms,
Nagoya Math. J. 240 (2020), 150-180.
MR4176743
- [10]
-
J. Kaczorowski and A. Perelli,
The standard twist of \(L\)-functions revisited,
arXiv:1911.10497, 2019.
- [11]
-
A. Perelli,
A survey of the Selberg class of \(L\)-functions, I,
Milan J. Math. 73 (2005), 19-52.
MR2175035
- [12]
-
A. Perelli,
A survey of the Selberg class of \(L\)-functions, II,
Riv. Mat. Univ. Parma (7) 3* (2004), 83-118.
MR2128842
- [13]
-
A. Perelli,
Non-linear twists of \(L\)-functions: a survey,
Milan J. Math. 78 (2010), 117-134.
MR2684775
- [14]
-
A. Perelli,
Converse theorems: from the Riemann zeta function to the Selberg class,
Boll. Unione Mat. Ital. 10 (2017), 29-53.
MR3615161
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