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In this paper, we consider the relationship between the zeros of degree zero functions from the extended Selberg class and their derivatives left of the critical line. Specifically, we prove Speiser’s result for such functions, that is that the function satisfies the Riemann Hypothesis if and only if its derivative has no zeros left of the critical line. This result allows us to give an easy to check sufficient condition for a degree zero function from the extended Selberg class to satisfy the Riemann hypothesis based on its coefficients. Additionally, we investigate families of such functions parameterized by a real variable and give conditions under which their zero trajectories leave the critical line. [ABSTRACT FROM AUTHOR]

We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number a ≠ 0 and a function from the Selberg class L , we prove a Riemann–von Mangoldt formula for the number of a -points of the Δ -factor of the functional equation of L and an analog of Landau's formula over these points. From the last formula we derive that the ordinates of these a -points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of L (s) taken at these points. [ABSTRACT FROM AUTHOR]

We investigate the order of exponential sums involving the coefficients of general L -functions satisfying a suitable functional equation and give some new estimates, including refining certain results in preceding works [X. Ren and Y. Ye, Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for G L m (ℤ) , Sci. China Math.58(10) (2015) 2105–2124; Y. Jiang and G. Lü, Oscillations of Fourier coefficients of Hecke–Maass forms and nonlinear exponential functions at primes, Funct. Approx. Comment. Math.57 (2017) 185–204]. [ABSTRACT FROM AUTHOR]

Let f be an arithmetic function and let S # denote the extended Selberg class. We denote by L (s) = ∑ n = 1 ∞ f (n) n s the Dirichlet series attached to f. The Laurent–Stieltjes constants of L (s) , which belongs to S # , are the coefficients of the Laurent expansion of L at its pole s = 1 . In this paper, we give an upper bound of these constants, which is a generalization of many known results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Levinson and Montgomery proved that the Riemann zeta-function ζ(s) and its derivative have approximately the same number of nonreal zeros left of the critical line. Spira showed that ζ ′ (1/2+it) = 0 implies that ζ(1/2+it) = 0. Here we obtain that in small areas located to the left of the critical line and near it the functions ζ(s) and ζ ′ (s) have the same number of zeros. We prove our result for more general zeta-functions from the extended Selberg class S. We also consider zero trajectories of a certain family of zeta-functions from S. [ABSTRACT FROM AUTHOR]

We study the uniqueness of functions in the extended Selberg class. It was shown in Ki (Adv Math 231, 2484-2490, ) that if for a nonzero complex number $$c$$ the inverse images $$L_1^{-1}(c)$$ and $$L_2^{-1}(c)$$ of two functions satisfying the same functional equation in the extended Selberg class are the same, then $$L_1(s)$$ and $$L_2(s)$$ are identical. Here we prove that this holds even without the assumption that they satisfy the same functional equation. [ABSTRACT FROM AUTHOR]