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Ade Irma Suriajaya

Publications and source records attributed to Ade Irma Suriajaya.

31 records · Page 2Linked to original sources
arXiv2021

Riemann-Type Functional Equations -- Julia Line and Counting Formulae --

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 $a\neq0$ and a function from the Selberg class $\mathcal{L}$, we prove a Riemann-von Mangoldt formula for the number of a-points of the $Δ$-factor of the functional equation of $\mathcal{L}$ 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 $\mathcal{L}(s)$ taken at these points.

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arXiv2020

A singular series average and the zeros of the Riemann zeta-function

We show that the Riesz mean of the singular series in the Goldbach and the Hardy-Littlewood prime-pair conjectures has an asymptotic formula with an error term that can be expressed as an explicit formula that depends on the zeros of the Riemann zeta-function. Unconditionally this error term can be shown to oscillate, while conditionally it can be shown to oscillate between sharp bounds.

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arXiv2020

Note on the number of zeros of $ζ^{(k)}(s)$

Assuming the Riemann hypothesis, we prove that $$ N_k(T) = \frac{T}{2π}\log \frac{T}{4πe} + O_k\left(\frac{\log{T}}{\log\log{T}}\right), $$ where $N_k(T)$ is the number of zeros of $ζ^{(k)}(s)$ in the region $0<\Im s\le T$. We further apply our method and obtain a zero counting formula for the derivative of Selberg zeta functions, improving earlier work of Luo.

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arXiv2020

Value-Distribution of the Riemann Zeta-Function along its Julia Lines

For an arbitrary complex number $a\neq 0$ we consider the distribution of values of the Riemann zeta-function $ζ$ at the $a$-points of the function $Δ$ which appears in the functional equation $ζ(s)=Δ(s)ζ(1-s)$. These $a$-points $δ_a$ are clustered around the critical line $1/2+i\mathbb{R}$ which happens to be a Julia line for the essential singularity of $ζ$ at infinity. We observe a remarkable average behaviour for the sequence of values $ζ(δ_a)$.

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arXiv2020

Dirichlet Series with Periodic Coefficients and their Value-Distribution Near the Critical Line

The class of Dirichlet series associated with a periodic arithmetical function $f$ includes the Riemann zeta-function as well as Dirichlet $L$-functions to residue class characters. We study the value-distribution of these Dirichlet series $L(s;f)$, resp. their analytic continuation in the neighborhood of the critical line (which is the abscissa of symmetry of the related Riemann-type functional equation). In particular, for a fixed complex number $a\neq 0$, we prove for an even or odd periodic $f$ the number of $a$-points of the $Δ$-factor of the functional equation, prove the existence of the mean-value of the values of $L(s;f)$ taken at these points, show that the ordinates of these $a$-points are uniformly distributed modulo one and apply this to show a discrete universality theorem.

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arXiv2018

The Values of the Riemann Zeta-Function on Discrete Sets

We study the values taken by the Riemann zeta-function $ζ$ on discrete sets. We show that infinite vertical arithmetic progressions are uniquely determined by the values of $ζ$ taken on this set. Moreover, we prove a joint discrete universality theorem for $ζ$ with respect to certain permutations of the set of positive integers. Finally, we study a generalization of the classical denseness theorems for $ζ$.

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arXiv2017

Zeros of the first derivative of Dirichlet $L$-functions

Yıldırım has classified zeros of the derivatives of Dirichlet $L$-functions into trivial zeros, nontrivial zeros and vagrant zeros. In this paper we remove the possibility of vagrant zeros for $L'(s,χ)$ when the conductors are large to some extent. Then we improve asymptotic formulas for the number of zeros of $L'(s,χ)$ in $\{s\in\mathbb{C}:\operatorname{Re}(s)>0, |\operatorname{Im}(s)|\leq T\}$. We also establish analogues of Speiser's theorem, which characterize the generalized Riemann hypothesis for $L(s,χ)$ in terms of zeros of $L'(s,χ)$, when the conductor is large.

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arXiv2016

An ergodic value distribution of certain meromorphic functions

We calculate a certain mean-value of meromorphic functions by using specific ergodic transformations, which we call affine Boolean transformations. We use Birkhoff's ergodic theorem to transform the mean-value into a computable integral which allows us to completely determine the mean-value of this ergodic type. As examples, we introduce some applications to zeta functions and $L$-functions. We also prove an equivalence of the Lindelöf hypothesis of the Riemann zeta function in terms of its certain ergodic value distribution associated with affine Boolean transformations.

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arXiv2016

Two estimates on the distribution of zeros of the first derivative of Dirichlet $L$-functions under the generalized Riemann hypothesis

Zeros of the Riemann zeta function and its derivatives have been studied by many mathematicians. Among, the number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, Akatsuka, and the author. Berndt, Levinson, and Montgomery investigated the unconditional case, while Akatsuka and the author gave sharper estimates under the truth of the Riemann hypothesis. In this paper, we prove similar results related to the first derivative of the Dirichlet $L$-functions associated with primitive Dirichlet characters under the assumption of the generalized Riemann hypothesis.

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arXiv2015

On the zeros of the $k$-th derivative of the Riemann zeta function under the Riemann hypothesis

The number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, and Akatsuka. Berndt, Levinson, and Montgomery investigated the general case, meanwhile Akatsuka gave sharper estimates for the first derivative of the Riemann zeta function under the truth of the Riemann hypothesis. In this paper, we generalize the results of Akatsuka to the $k$-th derivative (for positive integer $k$) of the Riemann zeta function.

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arXiv2013

On the zeros of the second derivative of the Riemann zeta function under the Riemann hypothesis

The number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, and Akatsuka. Berndt, Levinson, and Montgomery investigated the general case, meanwhile Akatsuka gave sharper estimates for the first derivative of the Riemann zeta function under the truth of the Riemann hypothesis. In this paper, we extend the results of Akatsuka to the second derivative of the Riemann zeta function.

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