Abstract
Let \(S(t,F):=\pi ^{-1}\arg L\big (\frac{1}{2}+it,F\big ),\) where F is a Hecke–Maass cusp form for \(\mathrm {SL}_3({\mathbb {Z}}).\) We establish an asymptotic formula for the spectral moments of S(t, F), and obtain several other results on S(t, F).
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References
Backlund, R.J.: Sur las zéros de la fonction \(\zeta (s)\) de Riemann. C. R. Acad. Sci. Paris 158, 1979–1981 (1914)
Backlund, R.J.: Über die Nullstellen der Riemannschen Zetafunktion. Acta Math. 41, 345–375 (1918)
Billingsley, P.: Probability and Measure, 3rd edn. Wiley Series in Probability and Mathematical Statistics, Wiley, New York (1995)
Buttcane, J., Zhou, F.: Plancherel distribution of satake parameters of Maass Cusp forms on \({{\rm GL}}_3\). International Mathematics Research Notices, to appear
Goldfeld, D.: Automorphic Forms and \(L\)-Functions for the Group \({{\rm GL}} (n,{\mathbb{R}})\), with an appendix by Kevin A. Broughan, vol. 99. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (2006)
Hejhal, D., Luo, W.: On a spectral analog of Selberg’s result on \(S(T)\). Int. Math. Res. Notices 3, 135–151 (1997)
Kim, H.H.: Functoriality for the exterior square of \({{\rm GL}}_4\) and the symmetric fourth of \({{\rm GL}}_2\). With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. J. Am. Math. Soc. 16(1), 139–183 (2003)
Littlewood, J.E.: On the zeros of the Riemann zeta-function. Proc. Camb. Philos. Soc. 2(24), 295–318 (1924)
MacDonald, I.G.: Symmetric Functions and Hall Polynomials. With Contributions by A. Zelevinsky, 2nd edn. Oxford Science Publications. The Clarendon Press, Oxford Mathematical Monographs. Oxford University Press, New York (1995)
von Mangoldt, H.: Zu Riemanns Abhandlung “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse”. J. Reine Angew. Math. 114, 255–305 (1895)
von Mangoldt, H.: Zur Verteilung der Nullstellen der Riemannschen Funktion \(\xi (t)\). Math. Ann. 60(1), 1–19 (1905)
Selberg, A.: On the Remainder in the Formula for \(N(T)\), the Number of Zeros of \(\zeta (s)\) in the strip \(0<t<T\). Avh. Norske Vid. Akad. Oslo. I.,. no. 1, p. 27; Collected Papers I, vol. 1989, pp. 179–203. Springer, Berlin (1944)
Selberg, A.: Contributions to the theory of the Riemann zeta-function. Arch. Math. Naturvid. 48(5), 89–155; Collected Papers I, vol. 1989, pp. 214–280. Springer, Berlin (1946)
Selberg, A.: Contributions to the theory of Dirichlet’s \(L\)-functions. Skr. Norske Vid. Akad. Oslo. I.,: (1946). no. 3, p. 62; Collected Papers I, vol. 1989, pp. 281–340. Springer, Berlin (1946)
Titchmarsh, E.C.: The zeros of the Riemann zeta-function. Proc. R. Soc. 151, 234–255 (1935)
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The authors would like to thank the referee for a careful reading and valuable comments.
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Sheng-Chi Liu was supported by a Grant (#344139) from the Simons Foundation.
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Liu, SC., Liu, S. A \(\mathrm {GL}_3\) analog of Selberg’s result on S(t). Ramanujan J 56, 163–181 (2021). https://doi.org/10.1007/s11139-020-00308-4
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DOI: https://doi.org/10.1007/s11139-020-00308-4