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Zeros and poles of Dirichlet series

Enrico Bombieri, Alberto Perelli (2001)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Under certain mild analytic assumptions one obtains a lower bound, essentially of order r , for the number of zeros and poles of a Dirichlet series in a disk of radius r . A more precise result is also obtained under more restrictive assumptions but still applying to a large class of Dirichlet series.

Zeros of eigenfunctions of some anharmonic oscillators

Alexandre Eremenko, Andrei Gabrielov, Boris Shapiro (2008)

Annales de l’institut Fourier

We study complex zeros of eigenfunctions of second order linear differential operators with real even polynomial potentials. For potentials of degree 4, we prove that all zeros of all eigenfunctions belong to the union of the real and imaginary axes. For potentials of degree 6, we classify eigenfunctions with finitely many zeros, and show that in this case too, all zeros are real or pure imaginary.

Zeros of solutions of certain higher order linear differential equations

Hong-Yan Xu, Cai-Feng Yi (2010)

Annales Polonici Mathematici

We investigate the exponent of convergence of the zero-sequence of solutions of the differential equation f ( k ) + a k - 1 ( z ) f ( k - 1 ) + ⋯ + a ₁ ( z ) f ' + D ( z ) f = 0 , (1) where D ( z ) = Q ₁ ( z ) e P ₁ ( z ) + Q ₂ ( z ) e P ₂ ( z ) + Q ₃ ( z ) e P ₃ ( z ) , P₁(z),P₂(z),P₃(z) are polynomials of degree n ≥ 1, Q₁(z),Q₂(z),Q₃(z), a j ( z ) (j=1,..., k-1) are entire functions of order less than n, and k ≥ 2.

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