Related papers: Failed attempt to disproof the Riemann Hypothesis
We study a general framework of distributional computational graphs: computational graphs whose inputs are probability distributions rather than point values. We analyze the discretization error that arises when these graphs are evaluated…
Riemann's hypothesis, formulated in 1859, concerns the location of the zeros of Riemann's Zeta function. The history of the Riemann hypothesis is well known. In 1859, the German mathematician B. Riemann presented a paper to the Berlin…
We construct a Beurling generalized number system satisfying the Riemann hypothesis and whose integer counting function displays extremal oscillation in the following sense. The prime counting function of this number system satisfies…
Online learning methods yield sequential regret bounds under minimal assumptions and provide in-expectation risk bounds for statistical learning. However, despite the apparent advantage of online guarantees over their statistical…
We analyze the computational efficiency of approximate Bayesian computation (ABC), which approximates a likelihood function by drawing pseudo-samples from the associated model. For the rejection sampling version of ABC, it is known that…
We address the problem of detecting deviations of binary sequence from randomness,which is very important for random number (RNG) and pseudorandom number generators (PRNG). Namely, we consider a null hypothesis $H_0$ that a given bit…
In this paper we study the convergence of online gradient descent algorithms in reproducing kernel Hilbert spaces (RKHSs) without regularization. We establish a sufficient condition and a necessary condition for the convergence of excess…
The assumption of real-number arithmetic, which is at the basis of conventional geometric algorithms, has been seriously challenged in recent years, since digital computers do not exhibit such capability. A geometric predicate usually…
This note describes the results of some tests of the ABC-PRC algorithm of Sissons et al. (PNAS, 2007), and demonstrates with a toy example that the method does not converge on the true posterior distribution.
The compensated quotient-difference (Compqd) algorithm is proposed along with some applications. The main motivation is based on the fact that the standard quotient-difference (qd) algorithm can be numerically unstable. The Compqd algorithm…
We point out that $R \equiv Br(b\rightarrow d \gamma)/Br(b\rightarrow s \gamma)$ is a sensitive probe of possible violation of CKM unitarity. We compute $R$ in a minimal extension of the Standard Model containing an additional isosinglet…
Numerical integrations in celestial mechanics often involve the repeated computation of a rotation with a constant angle. A direct evaluation of these rotations yields a linear drift of the distance to the origin. This is due to roundoff in…
We present a procedure for handling asymmetric errors. Many results in particle physics are presented as values with different positive and negative errors, and there is no consistent procedure for handling them. We consider the difference…
The finite Dirichlet series from the title are defined by the condition that they vanish at as many initial zeroes of the zeta function as possible. It turned out that such series can produce extremely good approximations to the values of…
This paper improves algorithms given in math.CO/0012036. Although the graph (digraph) becomes non-random as the algorithm proceeds, the probability for success stays the same. We also give examples.
As well known, the important hypothesis formulated by B.G. RIEMANN in 1859 states that all non-trivial zeroes of the Zeta function $Z(s)=\sum_{n=1}^{\infty } n^{-s}$ should fall on the Critical Line (C.L.) $Re(s)=\frac{1}{2}$.\\ Although…
In this paper, we define the geometric median of a probability measure on a Riemannian manifold, give its characterization and a natural condition to ensure its uniqueness. In order to calculate the median in practical cases, we also…
The validity of the Riemann Hypothesis (RH) on the location of the non-trivial zeros of the Riemann $\zeta$-function is directly related to the growth of the Mertens function $M(x) \,=\,\sum_{k=1}^x \mu(k)$, where $\mu(k)$ is the M\"{o}bius…
There are many formulations of problems that have been proven to be equivalent to the Riemann Hypothesis in modern mathematics. In this paper we look at the formulation of an inequality derived by Robin in 1984 that proves the Riemann…
Approximate Markov chain Monte Carlo (MCMC) offers the promise of more rapid sampling at the cost of more biased inference. Since standard MCMC diagnostics fail to detect these biases, researchers have developed computable Stein discrepancy…