On the sign changes of coefficients of general Dirichlet series

Pribitkin, Wladimir

doi:10.1090/S0002-9939-08-09296-4

On the sign changes of coefficients of general Dirichlet series
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by Wladimir de Azevedo Pribitkin PDF

Proc. Amer. Math. Soc. 136 (2008), 3089-3094 Request permission

Abstract:

Under what conditions do the (possibly complex) coefficients of a general Dirichlet series exhibit oscillatory behavior? In this work we invoke Laguerre’s Rule of Signs and Landau’s Theorem to provide a rather simple answer to this question. Furthermore, we explain how our result easily applies to a multitude of functions.

References

Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
Norman Bleistein and Richard A. Handelsman, Asymptotic expansions of integrals, 2nd ed., Dover Publications, Inc., New York, 1986. MR 863284
M. Aslam Chaudhry and Syed M. Zubair, On a class of incomplete gamma functions with applications, Chapman & Hall/CRC, Boca Raton, FL, 2002. MR 1887130
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Tables of Integral Transforms, 1, Bateman Manuscript Project, McGraw-Hill, New York, 1954.
G. H. Hardy and M. Riesz, The general theory of Dirichlet’s series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 18, Stechert-Hafner, Inc., New York, 1964. MR 0185094
Marvin Knopp, Winfried Kohnen, and Wladimir Pribitkin, On the signs of Fourier coefficients of cusp forms, Ramanujan J. 7 (2003), no. 1-3, 269–277. Rankin memorial issues. MR 2035806, DOI 10.1023/A:1026207515396
Winfried Kohnen, Sign changes of Hecke eigenvalues of Siegel cusp forms of genus two, Proc. Amer. Math. Soc. 135 (2007), no. 4, 997–999. MR 2262899, DOI 10.1090/S0002-9939-06-08570-4
E.N. Laguerre, Sur la Théorie des Équations Numériques, J. Math. Pures Appl. 9 (1883), 99-146 (also in Œuvres de Laguerre, v. I, pp. 3-47, Gauthier-Villars, Paris, 1898).
Edmund Landau, Über einen Satz von Tschebyschef, Math. Ann. 61 (1906), no. 4, 527–550 (German). MR 1511360, DOI 10.1007/BF01449495
George Pólya and Gabor Szegő, Problems and theorems in analysis. II, Classics in Mathematics, Springer-Verlag, Berlin, 1998. Theory of functions, zeros, polynomials, determinants, number theory, geometry; Translated from the German by C. E. Billigheimer; Reprint of the 1976 English translation. MR 1492448, DOI 10.1007/978-3-642-61905-2_{7}
J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216, DOI 10.1007/978-1-4684-9884-4