Zeros of the Higher-Order Derivatives of the Functions Belonging to the Extended Selberg Class
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Lithuanian Mathematical Journal
Zeros of the higher-order derivatives of the functions belonging to the extended Selberg class∗ Raivydas Šim˙enas Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania (e-mail: [email protected]) Received March 5, 2020; revised June 8, 2020
Abstract. We study the distribution of the zeros of the kth derivatives of the functions belonging to the extended Selberg class. We obtain the zero-free regions for these derivatives and a Riemann–von Mangoldt-type estimate of the count of their nontrivial zeros. MSC: 11M26, 11M41 Keywords: extended Selberg class, higher-order derivatives
1 Introduction 1.1
Historical background
The study of the derivatives of zeta functions was to a large extent inspired by the Speiser’s paper [9]. Speiser proved that the Riemann hypothesis is equivalent to the absence of zeros of the derivative of the Riemann zeta function to the left of the critical line. In 1970, Berndt [1] studied the number of zeros of the higher-order derivatives of the Riemann zeta function, and earlier in 1965, Spira [10] investigated the zero-free regions of the higher-order derivatives of the Riemann zeta function. Levinson and Montgomery [6] proved that, under certain conditions, the Riemann zeta function and its derivative have approximately the same number of zeros to the left of the critical line. They also studied the zeros of the higher-order derivatives of the Riemann zeta function, and assuming the Riemann hypothesis, they proved that there are a finite number of such zeros in the critical strip. Later on, Yıldırım [15] considered the number of zeros of the first- and higher-order derivatives of the Dirichlet L-functions. He obtained zero-free regions and an estimate of the number of zeros for these functions. Garunkštis and Šim˙enas [2] studied the relationship between the zeros of the functions belonging to the extended Selberg class and of their first-order derivatives. Our result was that the numbers of such zeros to the left of the critical line are approximately the same. In addition, under certain conditions, there exists a sequence Tj → ∞ as j → ∞ such that the number of zeros of a given function F belonging to the extended Selberg class and of F coincide in the region of the complex plane bounded by the critical line on the right, some ∗
This project has received funding from European Social Fund (project No. 09.3.3-LMT-K-712-02-0088) under grant agreement with the Research Council of Lithuania (LMTLT).
c 2020 Springer Science+Business Media, LLC 0363-1672/20/6004-0001
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line t = T1 on the bottom, and t = Tj on the top. Here, as usual, s = σ + it is a complex variable. Our unconditional result was that the numbers of such zeros differ by O(log T ). Our investigation differs from Speiser [9], Berndt [1], Spira [10, 11], and Levinson and Montgomery [6] in that we study functions that may have some zeros to the left of the critical line, as the Davenport–Heilb
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