Nonlinear exponential twists of the Liouville function

• Autorzy: Qingfeng Sun
• Języki publikacji: EN

Abstrakty

EN

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).

Słowa kluczowe

EN

Liouville function; Nonlinear exponential sums; Vaughan’s identity

Kategorie tematyczne

• 11M41: Other Dirichlet series and zeta functions
• 11L07: Estimates on exponential sums

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2011
• Tom: 9
• Numer: 2
• Strony: 328-337

Daty

wydano

2011-04-01

online

2011-02-18

Twórcy

autor

Qingfeng Sun

• Shandong University at Weihai

Bibliografia

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• [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
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• [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] 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
• [8] Titchmarsh E.C., The Theory of the Riemann Zeta-Function, 2nd ed., University Press, New York, 1986
• [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
• [10] Zhao L.Y., Oscillations of Hecke eigenvalues at shifted primes, Rev. Mat. Iberoam., 2006, 22(1), 323–337

Identyfikatory

DOI

10.2478/s11533-010-0092-6