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We show that the sign constancy for the values of certain weighted summatory functions of the von Mangoldt function implies the Riemann hypothesis or the generalized Riemann hypothesis for Dirichlet $L$-functions. While such sign constancy is challenging to establish individually, we prove that the summatory functions under study have constant signs on average., Comment: 19 pages

We study the $M$-functions, which describe the limit theorem for the value-distributions of the secondary main terms in the asymptotic formulas for the summatory functions of the Goldbach counting function. One of the new aspects is a sufficient condition for the Riemann hypothesis provided by some formulas of the $M$-functions, which was a necessary condition in previous work. The other new aspect is the relation between the secondary main terms and the screw functions, which provides another necessary and sufficient condition for the Riemann hypothesis. We study such $M$-functions and screw functions in generalized settings by axiomatizing them., Comment: 19 pages

We present several analytic theories and their relations to each other related to screw functions, taking the screw function of the simplest screw line as an example. Note that this article does not contain any new results., Comment: 34 pages. This article is a kind of survey and does not contain new results

We show that central zeros of $L$-functions in the Selberg class have a probabilistic interpretation by stating an equivalence condition of the Riemann hypothesis for the $L$-functions in terms of infinitely divisible distributions., Comment: 6 pages

For the Riemann zeta-function, we introduce a function such that it is a characteristic function of an infinitely divisible distribution on the real line if and only if the Riemann Hypothesis is true., Comment: 5 pages. To appear in Statistics and Probability Letters

It is known that the nonnegativity of Li coefficients is a necessary and sufficient condition for the Riemann hypothesis. We show that it is a necessary and sufficient condition for the Riemann hypothesis that all Li coefficients are norms of certain concrete functions on the real line. Such conditional formulas for Li coefficients are understood as a kind of Weil's criterion for the Riemann hypothesis., Comment: 13 pages

We study the Hilbert space obtained by completing the space of all smooth and compactly supported functions on the real line with respect to the hermitian form arising from the Weil distribution under the Riemann hypothesis. It turns out that this Hilbert space is isomorphic to a de Branges space by a composition of the Fourier transform and a simple map.This result is applied to state a new equivalence condition for the Riemann hypothesis in a series of equalities., Comment: 19 pages. This paper has been merged with arxiv:2209.04658 in the latest version

We introduce screw functions for Dirichlet series in the extended Selberg class. Then we prove that the Grand Riemann Hypothesis for a member of the extended Selberg class is equivalent to the nonpositivity of the corresponding screw function., Comment: 6 pages

We investigate the screw line corresponding to the screw function associated with the Riemann zeta-function under the Riemann hypothesis and derive three necessary and sufficient conditions for the Riemann hypothesis as applications. One of them explains the non-negativity of the Weil distribution by means of the norm., Comment: 9 pages. One main result was added in the 3rd revision

We introduce a screw function corresponding to the Riemann zeta-function and study its properties from various aspects. Typical results are several equivalent conditions for the Riemann hypothesis in terms of the screw function. One of them can be considered an analog of so-called Weil's positivity or Li's criterion. In addition, we prove a few partial but unconditional results for such equivalents., Comment: 32 pages. The proof of Theorem 1.4 has been revised, and the organization of the paper has been significantly revised in the latest version

We solve direct and inverse problems for two-dimensional (quasi) canonical systems related to exponential polynomials of a specific but sufficiently general type. The approach to the inverse problem in this paper provides an interpretation of the matrices and their determinants in the classical Schur-Cohn test for polynomials in terms of Hamiltonians of canonical system., Comment: 30 pages

We review the EfficientQA competition from NeurIPS 2020. The competition focused on open-domain question answering (QA), where systems take natural language questions as input and return natural language answers. The aim of the competition was to build systems that can predict correct answers while also satisfying strict on-disk memory budgets. These memory budgets were designed to encourage contestants to explore the trade-off between storing retrieval corpora or the parameters of learned models. In this report, we describe the motivation and organization of the competition, review the best submissions, and analyze system predictions to inform a discussion of evaluation for open-domain QA., Comment: 26 pages; Published in Proceedings of Machine Learning Research (PMLR), NeurIPS 2020 Competition and Demonstration Track

We present a method to construct a chain of reproducing kernel Hilbert spaces controlled by a first-order system of differential equations from a given unimodular function satisfying several conditions. One of the applications of that method is a conditional but richly general solution to the inverse problem of recovering the structure Hamiltonian from a given de Branges space., Comment: 31 pages, 0 figures. To appear in Annales de l'Institut Fourier

Hamiltonians are 2-by-2 positive semidefinite real symmetric matrix-valued functions satisfying certain conditions. In this paper, we solve the inverse problem for which recovers a Hamiltonian from the solution of a first-order system attached to a given Hamiltonian, consisting of ordinary differential equations parametrized by a set of complex numbers, under certain conditions for the solutions. This inverse problem is a generalization of the inverse problem for two-dimensional canonical systems., Comment: 16 pages, 0 figures

In this paper we have two issues coming from the same background. The first one is to describe a certain ratio of Fredholm determinants of integral operators arising from the Riemann zeta function by using the solution of a single integral equation. The second one is to introduce a new integral operator arising from the Riemann zeta function and to study its basic analytic properties., Comment: 10 pages

We establish a new equivalent condition for the Grand Riemann Hypothesis for L-functions in a wide subclass of the Selberg class in terms of canonical systems of differential equations. A canonical system is determined by a real symmetric matrixvalued function called a Hamiltonian. To establish the equivalent condition, we use an inverse problem for canonical systems of a special type., Comment: The paper has been significantly revised, because a part of the old version was published as the reference [45] of the latest version

We show that the density functions of nearest neighbor spacing distributions for zeros of the real or imaginary part of the Riemann xi-function on vertical lines are described by the M-function which is appeared in value distributions of the logarithmic derivative of the Riemann zeta-function on vertical lines., Comment: 15 pages

A canonical system is a kind of first-order system of ordinary differential equations on an interval of the real line parametrized by complex numbers. It is known that any solution of a canonical system generates an entire function of the Hermite-Biehler class. In this paper, we deal with the inverse problem to recover a canonical system from a given entire function of the Hermite-Biehler class satisfying appropriate conditions. This type inverse problem was solved by de Branges in 1960s. However his results are often not enough to investigate a Hamiltonian of recovered canonical system. In this paper, we present an explicit way to recover a Hamiltonian from a given exponential polynomial belonging to the Hermite-Biehler class. After that, we apply it to study distributions of roots of self-reciprocal polynomials., Comment: 47 pages; an upgraded version of arXiv:1211.2953; the latest version is a revision after publication

We establish a necessary and sufficient condition for all zeros of a self-reciprocal polynomial to lie on the unit circle. Moreover, we relate the necessary and sufficient condition with a canonical system of linear differential equations (in the sense of de Branges). This relationship enable us to understand that the property of a self-reciprocal polynomial having only zeros on the unit circle is equivalent to the positive semidefiniteness of Hamiltonians of corresponding canonical systems., Comment: 28 pages; v2 29 pages, revision of Sec.8, typos corrected, refs. added

We state some sufficient or equivalent conditions to GRH of general L-functions in terms of monotonicity of certain weighted summatory functions., Comment: 13 pages

This paper has two main results, which relate to a criteria for the Riemann hypothesis via the family of functions $\Theta_\omega(z)=\xi(1/2-\omega-iz)/\xi(1/2+\omega-iz)$, where $\omega>0$ is a real parameter and $\xi(s)$ is the Riemann xi-function. The first main result is necessary and sufficient conditions for $\Theta_\omega$ to be a meromorphic inner function in the upper half-plane. It is related to the Riemann hypothesis directly whether $\Theta_\omega$ is a meromorphic inner function. In comparison with this, a relation of the Riemann hypothesis and the second main result is indirect. It relates to the theory of de Branges, which associates a meromorphic inner function and a canonical system of linear differential equations (in the sense of de Branges). As the second main result, the canonical system associated with $\Theta_\omega$ is constructed explicitly and unconditionally under the restriction of the parameter $\omega >1$ by applying a method of J.-F. Burnol in his recent work on the gamma function to the Riemann xi-function. If such construction is extended to all $\omega > 0$ unconditionally, we get a criterion for the Riemann hypothesis in terms of a family of canonical systems parametrized by $\omega>0$, which explains the validity of the Riemann hypothesis as positive semidefiniteness of the corresponding family of Hamiltonian matrices., Comment: 28 pages, draft of a paper which will be published in a volume of RIMS Kokyuroku Bessatsu Series

We prove that all but finitely many zeros of Weng's zeta function for a Chevalley group defined over $\Q$ are simple and on the critical line under some reasonable geometric hypothesis., Comment: minor typo corrections in Introduction