Nonlinear exponential twists of the Liouville function

Qingfeng Sun

Open Mathematics (2011)

  • Volume: 9, Issue: 2, page 328-337
  • ISSN: 2391-5455

Abstract

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Let λ(n) be the Liouville function. We find a nontrivial upper bound for the sum ∑ X ⩽ n ⩽ 2 X λ ( n ) e 2 π i α n , 0 ≠ α ∈ ℝ The main tool we use is Vaughan’s identity for λ(n).

How to cite

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Qingfeng Sun. "Nonlinear exponential twists of the Liouville function." Open Mathematics 9.2 (2011): 328-337. <http://eudml.org/doc/269447>.

@article{QingfengSun2011,
abstract = {Let λ(n) be the Liouville function. We find a nontrivial upper bound for the sum \[ \sum \limits \_\{X \leqslant n \leqslant 2X\} \{\lambda (n)e^\{2\pi i\alpha \sqrt\{n\} \} \} ,0 \ne \alpha \in \mathbb \{R\} \] The main tool we use is Vaughan’s identity for λ(n).},
author = {Qingfeng Sun},
journal = {Open Mathematics},
keywords = {Liouville function; Nonlinear exponential sums; Vaughan’s identity; nonlinear exponential sums; Vaughan's identity},
language = {eng},
number = {2},
pages = {328-337},
title = {Nonlinear exponential twists of the Liouville function},
url = {http://eudml.org/doc/269447},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Qingfeng Sun
TI - Nonlinear exponential twists of the Liouville function
JO - Open Mathematics
PY - 2011
VL - 9
IS - 2
SP - 328
EP - 337
AB - Let λ(n) be the Liouville function. We find a nontrivial upper bound for the sum \[ \sum \limits _{X \leqslant n \leqslant 2X} {\lambda (n)e^{2\pi i\alpha \sqrt{n} } } ,0 \ne \alpha \in \mathbb {R} \] The main tool we use is Vaughan’s identity for λ(n).
LA - eng
KW - Liouville function; Nonlinear exponential sums; Vaughan’s identity; nonlinear exponential sums; Vaughan's identity
UR - http://eudml.org/doc/269447
ER -

References

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  4. [4] Murty M.R., Sankaranarayanan A., Averages of exponential twists of the Liouville function, Forum Math., 2002, 14(2), 273–291 http://dx.doi.org/10.1515/form.2002.012 Zbl1012.11076
  5. [5] Pi Q.H., Sun Q.F., Oscillations of cusp form coefficients in exponential sums, Ramanujan J., 2010, 21(1), 53–64 http://dx.doi.org/10.1007/s11139-008-9151-z Zbl1245.11057
  6. [6] Ren X.M., Vinogradov’s exponential sums over primes, Acta Arith., 2006, 124(3), 269–285 http://dx.doi.org/10.4064/aa124-3-5 
  7. [7] Sun Q.F., On cusp form coefficients in nonlinear exponential sums, Q. J. Math., 2010, 61(3), 363–372 http://dx.doi.org/10.1093/qmath/hap008 Zbl1245.11059
  8. [8] Titchmarsh E.C., The Theory of the Riemann Zeta-Function, 2nd ed., University Press, New York, 1986 Zbl0601.10026
  9. [9] Vinogradov I.M., Special Variants of the Method of Trigonometric Sums, Nauka, Moscow, 1976 (in Russian), English translation in: Vinogradov I.M., Selected Works, Springer, Berlin, 1985 Zbl0429.10023
  10. [10] Zhao L.Y., Oscillations of Hecke eigenvalues at shifted primes, Rev. Mat. Iberoam., 2006, 22(1), 323–337 Zbl1170.11024

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