## Functiones et Approximatio Commentarii Mathematici

### On sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$

#### Abstract

We derive a truncated Voronoi identity for rationally additively twisted sums of Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$, and as an application obtain a pointwise estimate and a second moment estimate for the sums in question.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 57, Number 2 (2017), 255-275.

Dates
First available in Project Euclid: 28 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.facm/1490688027

Digital Object Identifier
doi:10.7169/facm/1632

Mathematical Reviews number (MathSciNet)
MR3732898

Zentralblatt MATH identifier
06864174

#### Citation

Jääsaari, Jesse; Vesalainen, Esa V. On sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$. Funct. Approx. Comment. Math. 57 (2017), no. 2, 255--275. doi:10.7169/facm/1632. https://projecteuclid.org/euclid.facm/1490688027

#### References

• A. Booker, A test for identifying Fourier coefficients of automorphic forms and application to Kloosterman sums, Experiment. Math. 9 (2000), 571–581.
• A. Booker, Numerical tests of modularity, J. Ramanujan Math. Soc. 20 (2005), 283–339.
• J. Brüdern, Einführung in die analytische Zahlentheorie, Springer, 1995.
• H. Cramér, Über zwei Sätze von Herrn G. H. Hardy, Math. Z. 15 (1922), 200–210.
• K. Chandrasekharan, and R. Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. of Math. (2) 76 (1962), 93–136.
• K. Chandrasekharan, R. Narasimhan, On the mean value of the error term for a class of arithmetical functions, Acta Math. 112 (1964), 41–67.
• A.-M. Ernvall-Hytönen, On the error term in the approximate functional equation for exponential sums related to cusp forms, Int. J. Number Theory 4 (2008), 747–756.
• A.-M. Ernvall-Hytönen and K. Karppinen, On short exponential sums involving Fourier coefficients of holomorphic cusp Forms, Int. Math. Res. Not. IMRN, article ID rnn022 (2008), 1–44.
• A.-M. Ernvall-Hytönen, J. Jääsaari, and E.V. Vesalainen, Resonances and $\Omega$-results for exponential sums related to Maass forms for $\mathrm{SL}(n,\mathbb Z)$, J. Number Theory 153 (2015), 135–157.
• J.B. Friedlander and H. Iwaniec, Summation formulae for coefficients of $L$-functions, Canad. J. Math. 57 (2005), 494–505.
• D. Godber, Additive twists of Fourier coefficients of modular forms, J. Number Theory 133 (2013), 83–104.
• D. Goldfeld,Automorphic Forms and $L$-Functions for the Group $\mathrm{SL}(n,\mathbb Z)$, Cambridge Studies in Advanced Mathematics, 99, Cambridge University Press, 2006.
• D. Goldfeld and X. Li, Voronoi formulas on $\mathrm{GL}(n)$, Int. Math. Res. Not., article ID 86295 (2006), 1–25.
• D. Goldfeld and J. Sengupta, First moment of Fourier coefficients of $\mathrm{GL}(n)$ cusp forms, J. Number Theory 161 (2016), 435–443.
• A. Ivić, The Riemann Zeta-Function: Theory and Applications, Dover Publications, 2003.
• A. Ivić and W. Zhai, Higher moments of the error term in the divisor problem, Math. Notes 88 (2010), 338–346.
• J. Jääsaari and E.V. Vesalainen, Exponential sums related to Maass forms, arXiv:1409.7235.
• L. Ji (ed.), Geometry and Analysis, No. 2, Advanced Lectures in Mathematics 18, International Press, 2011.
• M. Jutila, On exponential sums involving the divisor function, J. Reine Angew. Math. 355 (1985), 173–190.
• M. Jutila, On exponential sums involving the Ramanujan function, Proc. Indian Acad. Sci. 97 (1987), 157–166.
• M. Jutila, Lectures on a Method in the Theory of Exponential Sums, Lectures on Mathematics 80, Tata Institute of Fundamental Research, 1987.
• H.H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures, J. Amer. Math. Soc. 16 (2003), 175–183.
• N.N. Lebedev, Special Functions & their Applications, Dover Publications, 1972.
• X. Li and M.P. Young, Additive twists of Fourier coefficients of symmetric-square lifts, J. Number Theory 132 (2012), 1626–1640.
• G. Lü, On sums involving coefficients of automorphic $L$-functions, Proc. Amer. Math. Soc. 137 (2009), 2879–2887.
• G. Lü, On averages of Fourier coefficients of Maass cusp forms, Arch. Math. (Basel) 100 (2013), 255–265.
• J. Meher, On sums of Fourier coefficients of automorphic forms for $\mathrm{GL}_r$, arXiv:1412.8567.
• T. Meurman, On exponential sums involving the Fourier coefficients of Maass wave forms, J. Reine Angew. Math. 384 (1988), 192–207.
• S.D. Miller, Cancellation in additively twisted sums on $\mathrm{GL}(n)$, Amer. J. Math. 128 (2006), 699–729.
• S.D. Miller and W. Schmid, Automorphic distributions, $L$-functions, and Voronoi summation for $\mathrm{GL}(3)$, Ann. of Math. 164 (2006), 423–488.
• S.D. Miller and W. Schmid, A general Voronoi summation formula for $\mathrm{GL}(n,\mathbb Z)$, in [1632:Ji?], 173–224.
• X. Ren and Y. Ye, Sums of Fourier coefficients of a Maass form for $\mathrm{SL}_3(\mathbb Z)$ twisted by exponential functions, Forum Math. 26 (2014), 221–238.
• E.C. Titchmarsh, The Theory of the Riemann Zeta-function, second edition revised by D.R. Heath-Brown, Oxford University Press, 1986.
• E.V. Vesalainen, Moments and oscillations of exponential sums related to cusp forms, preprint, arXiv:1402.2746, to appear in Math. Proc. Cambridge Philos. Soc.
• J.R. Wilton, A note on Ramanujan's function $\tau(n)$, Math. Proc. Cambridge Philos. Soc. 25 (1929), 121–129.