##
recently archived material
[Items are added to the top of this list as they are archived elsewhere.]
C.M. Bender, D.C. Brody and M.P. Müller, "Hamiltonian for the zeros of the Riemann zeta function", *Phys. Rev. Lett.* **118** (2017) 130201
[abstract:] "A Hamiltonian operator $\hat{H}$ is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical limit of $\hat{H}$ is $2xp$, which is consistent with the Berry–Keating conjecture. While $\hat{H}$ is not Hermitian in the conventional sense, $i\hat{H}$ is $\mathcal{PT}$ symmetric with a broken $\mathcal{PT}$ symmetry, thus allowing for the possibility that all eigenvalues of $\hat{H}$ are real. A heuristic analysis is presented for the construction of the metric operator to define an inner-product space, on which the Hamiltonian is Hermitian. If the analysis presented here can be made rigorous to show that $\hat{H}$ is manifestly self-adjoint, then this implies that the Riemann hypothesis holds true."
N. Wolchover, "Physicists attack math's $1,000,000 question", *Quanta* (4 April 2017) [somewhat misleading popular science article about the publication by Bender, *et al.*]
J.V. Bellissard, "Comment on "Hamiltonian for the zeros of the Riemann zeta function"" (preprint, 04/2017)
[abstract:] "This comment about the article "Hamiltonian for the Zeros of the Riemann Zeta Function", by C. M. Bender, D. C. Brody, and M. P. Müller, published recently in *Phys. Rev. Lett.* (*Phys. Rev. Lett.*, **118**, 130201, (2017)) gives arguments showing that the strategy proposed by the authors to prove the Riemann Hypothesis, does not actually work."
[added 17th April 2017]
Conference: "Modular Forms and Quantum Knot Invariants" (Banff, AB, 11-16 March 2018)
"The goal of this intense five-day workshop is to bring together international experts and young researchers in low-dimensional topology, number theory, string theory, quantum physics, algebraic geometry, conformal field theory, special functions and automorphic forms to discuss new developments and investigate potential directions for future research at the crossroads of modular forms and quantum knot invariants."
[added 17th April 2017]
Y. Pan, "How to measure the canonical commutation relation $[\hat{x},\hat{p}] = i\hbar$? in quantum mechanics with weak measurement?" (preprint 02/2017)
[abstract:] "The quantum weak value draws many attentions recently from theoretical curiosity to experimental applications. Now we design an unusual weak measuring procedure as the pre-selection, mid-selection and post-selection to study the correlation function of two weak values, which we called the weak correlation function. In this paper, we proposed an weak measurement experiment to measure the canonical commutator $[\hat{x},\hat{p}] = i\hbar$? in quantum mechanics. Furthurmore, we found the intriguing equivalence between the canonical commutation relation and Riemann hypothesis, and then obtained the weak value of nontrivial Riemann zeros. Finally, as an nontrivial example of weak correlations, we also passed successfully a testing on the (anti-)commutators of Pauli operators, which followed the experimental setup of the landmark paper of Aharonov, et al. in 1988. Our proposed experiments could hopefully test the fundamental canonical relationship in quantum worlds and trigger more testing experiments on weak correlations."
[added 17th April 2017]
A. Schwarz, V. Vologodsky and J. Walcher, "Integrality of framing and geometric origin of 2-functions" (preprint 02/2017)
[abstract:] "We say that a formal power series $\sum a_nz^n$ with rational coefficients is a 2-function if the numerator of the fraction $a_{n/p} - p^2a_n$ is divisible by $p^2$ for every prime number $p$. One can prove that 2-functions with rational coefficients appear as building block of BPS generating functions in topological string theory. Using the Frobenius map we define 2-functions with coefficients in algebraic number fields. We establish two results pertaining to these functions. First, we show that the class of 2-functions is closed under the so-called framing operation (related to compositional inverse of power series). Second, we show that 2-functions arise naturally in geometry as $q$-expansion of the truncated normal function associated with an algebraic cycle extending a degenerating family of Calabi-Yau 3-folds."
[added 17th April 2017]
P. Roggero, M. Nardelli and F. Di Noto, "Sum of the reciprocals of famous series: Mathematical connections with some sectors of theoretical physics and string theory" (preprint 01/2017)
[abstract:] "In this paper it has been calculated the sums of the reciprocals of famous series. The sum of the reciprocals gives fundamental information on these series. The higher this sum and larger numbers there are in series and vice versa. Furthermore we understand also what is the growth factor of the series and that there is a clear link between the sums of the reciprocal and the "intrinsic nature" of the series. We have described also some mathematical connections with some sectors of theoretical physics and string theory."
[added 7th January 2017]
School and Workshop on Modular Forms and Black Holes, 5–14 January 2017, National Institute of Science Education and Research, Bhubaneswar, India
[added 15th December 2016]
F. Murtagh, Hierarchical matching and regression with application to photometric redshift estimation" (preprint, 12/2016)
[abstract:] "This work emphasizes that heterogeneity, diversity, discontinuity, and discreteness in data is to be exploited in classification and regression problems. A global a priori model may not be desirable. For data analytics in cosmology, this is motivated by the variety of cosmological objects such as elliptical, spiral, active, and merging galaxies at a wide range of redshifts. Our aim is matching and similarity-based analytics that takes account of discrete relationships in the data. The information structure of the data is represented by a hierarchy or tree where the branch structure, rather than just the proximity, is important. The representation is related to *p*-adic number theory. The clustering or binning of the data values, related to the precision of the measurements, has a central role in this methodology. If used for regression, our approach is a method of cluster-wise regression, generalizing nearest neighbour regression. Both to exemplify this analytics approach, and to demonstrate computational benefits, we address the well-known photometric redshift or 'photo-z' problem, seeking to match Sloan Digital Sky Survey (SDSS) spectroscopic and photometric redshifts."
[added 15th December 2016]
R. Sasaki, Symmetric Morse potential is exactly solvable" (preprint, 11/2016)
[abstract:] "Morse potential $V_M(x)= g^2\exp (2x)-g(2h+1)\exp(x)$ is defined on the full
line, $-\infty<x<\infty$ and it defines an exactly solvable 1-d quantum
mechanical system with finitely many discrete eigenstates. By taking its right
half $0\le x<\infty$ and glueing it with the left half of its mirror image
$V_M(-x)$, $-\infty<x\le0$, the symmetric Morse potential $V(x)= g^2\exp
(2|x|)-g(2h+1)\exp(|x|)$ is obtained. The quantum mechanical system of this
piecewise analytic potential has infinitely many discrete eigenstates with the
corresponding eigenfunctions given by the Whittaker W function. The eigenvalues
are the square of the zeros of the Whittaker function $W_{k,\nu}(x)$ and its
linear combination with $W'_{k,\nu}(x)$ as a function of $\nu$ with fixed $k$
and $x$. This quantum mechanical system seems to offer an interesting example
for discussing the Hilbert--P\'olya conjecture on the pure imaginary zeros of
Riemann zeta function on Re$(s)=\tfrac12$."
[added 15th December 2016]
V. Balasubramanian, J.R. Fliss, R.G. Leigh and O. Parrikar,
Multi-boundary entanglement in Chern–Simons theory and link invariants" (preprint 11/2016)
[abstract:] "We consider Chern–Simons theory for gauge group $G$ at level $k$ on 3-manifolds $M_n$ with boundary consisting of $n$ topologically linked tori.
The Euclidean path integral on $M_n$ defines a quantum state on the boundary,
in the $n$-fold tensor product of the torus Hilbert space. We focus on the case
where $M_n$ is the link-complement of some $n$-component link inside the
three-sphere $S^3$. The entanglement entropies of the resulting states define
new, framing-independent link invariants which are sensitive to the topology of
the chosen link. For the Abelian theory at level $k$ ($G= U(1)_k$) we give a
general formula for the entanglement entropy associated to an arbitrary
$(m|n-m)$ partition of a generic $n$-component link into sub-links. The formula
involves the number of solutions to certain Diophantine equations with
coefficients related to the Gauss linking numbers (mod $k$) between the two
sublinks. This formula connects simple concepts in quantum information theory,
knot theory, and number theory, and shows that entanglement entropy between
sublinks vanishes if and only if they have zero Gauss linking (mod $k$). For $G
= SU(2)_k$, we study various two and three component links. We show that the
2-component Hopf link is maximally entangled, and hence analogous to a Bell
pair, and that the Whitehead link, which has zero Gauss linking, nevertheless
has entanglement entropy. Finally, we show that the Borromean rings have a
"W-like" entanglement structure (i.e., tracing out one torus does not lead to a
separable state), and give examples of other 3-component links which have
"GHZ-like" entanglement (i.e., tracing out one torus does lead to a separable
state)."
[added 15th December 2016]
J. Berra-Montiel and A. Molgado, "Polymeric quantum mechanics and the zeros of the Riemann zeta function" (preprint, 10/2016)
[abstract:] "We analyze the Berry–Keating model and the Sierra and Rodríguez–Laguna Hamiltonian within the polymeric quantization formalism. By using the polymer representation, we obtain for both models, the associated polymeric quantum Hamiltonians and the corresponding stationary wave functions. The self-adjointness condition provide a proper domain for the Hamiltonian operator and the energy spectrum, which turned out to be dependent on an introduced scale parameter. By performing a counting of semiclassical states, we prove that the polymer representation reproduces the smooth part of the Riemann–von Mangoldt formula, and introduces a correction depending on the energy and the scale parameter, which resembles the fluctuation behavior of the Riemann zeros."
[added 15th December 2016]
L. Albert and M. K.-H. Kiessling, "Order and Chaos in some deterministic infinite trigonometric products" (preprint, 09/2016)
[abstract:] "In this paper it is proved that $\prod_{n=1}^\infty
\left[\frac23+\frac13\cos\left(\frac{x}{n^{2}}\right)\right] = e^{- C
\,\sqrt{|x|} +\varepsilon(|x|)},$ with $|\varepsilon(|x|)| \leq K |x|^{1/3}$
for some $K>0$, and with $ C= \int\frac{\sin\xi^2}{2+\cos\xi^2}{\rm{d}}\xi;$
numerically, $C = 0.319905585... \sqrt{\pi}$. As a corollary this confirms a
surmise of Benoit Cloitre. The $O\big(|x|^{1/3}\big)$ error bound is
empirically found to be accurate for moderately sized $|x|$ but not for larger
$|x|$. This difference $\varepsilon(|x|)$ between Cloitre's $\log
\prod_{n\geq1}\left[\frac23 +\frac13\cos\left(\frac{x}{n^{2}}\right)\right]$
and its regular trend $-C\sqrt{|x|}$, although deterministic, appears to be an
"empirically unpredictable" function. A probabilistic investigation of this
phenomenon is carried out, proving that Cloitre's trigonometric product is the
characteristic function of a simple random walk on the real interval
$(-\zeta(2),\zeta(2))$, where $\zeta$ is Riemann's zeta function, in fact, this
random walk is a 'random Riemann-$\zeta$ function with argument 2.' A few
related random walks are studied empirically and compared with their
theoretical distributions and trend distributions. The paper closes with a
paradoxical random-walk scenario and a remark on the Riemann hypothesis."
[added 15th December 2016]
T.N. Palmer, "*p*-Adic distance, finite precision and emergent superdeterminism: A number-theoretic consistent-histories approach to local quantum realism" (preprint, 09/2016)
[abstract:] "Although the notion of superdeterminism can, in principle, account for the violation of the Bell inequalities, this potential explanation has been roundly rejected by the quantum foundations community. The arguments for rejection, one of the most substantive coming from Bell himself, are critically reviewed. In particular, analysis of Bell's argument reveals an implicit unwarranted assumption: that the Euclidean metric is the appropriate yardstick for measuring distances in state space. Bell's argument is largely negated if this yardstick is instead based on the alternative *p*-adic metric. Such a metric, common in number theory, arises naturally when describing chaotic systems which evolve precisely on self-similar invariant sets in their state space. A locally-causal realistic model of quantum entanglement is developed, based on the premise that the laws of physics ultimately derive from an invariant-set geometry in the state space of a deterministic quasi-cyclic mono-universe. Based on this, the notion of a complex Hilbert vector is reinterpreted in terms of an uncertain selection from a finite sample space of states, leading to a novel form of 'consistent histories' based on number-theoretic properties of the transcendental cosine function. This leads to novel realistic interpretations of position/momentum non-commutativity, EPR, the Bell Theorem and the Tsirelson bound. In this inherently holistic theory – neither conspiratorial, retrocausal, fine tuned nor nonlocal – superdeterminism is not invoked by fiat but is emergent from these 'consistent histories' number-theoretic constraints. Invariant set theory provides new perspectives on many of the contemporary problems at the interface of quantum and gravitational physics, and, if correct, may signal the end of particle physics beyond the Standard Model."
[added 15th December 2016]
A. Sowa, "Encoding spatial data into quantum observables" (preprint 09/2016)
[abstract:] "The focus of this work is a correspondence between the Hilbert space operators on one hand, and doubly periodic generalized functions on the other. The linear map that implements it, referred to as the Q-transform, enables a direct application of the classical Harmonic Analysis in a study of quantum systems. In particular, the Q-transform makes it possible to reinterpret the dynamic of a quantum observable as a (typically nonlocal) dynamic of a classical observable. From this point of view we carry out an analysis of an open quantum system whose dynamics are governed by an asymptotically harmonic Hamiltonian and compact type Lindblad operators. It is established that the initial value problem of the equivalent nonlocal but classical evolution is well posed in the appropriately chosen Sobolev spaces. The second set of results pertains to a generalization of the basic Q-transform and highlights a certain type of asymptotic redundancy. This phenomenon, referred to as the broadband redundancy, is a consequence of a well-known property of the zeros of the Riemann zeta function, namely, the uniform distribution modulo one of their ordinates. Its relevance to the analysis of quantum dynamics is only a special instance of its utility in harmonic analysis in general. It remains to be seen if the phenomenon is significant also in the physical sense, but it appears well-justified—in particular, by the results presented here—to pose such a question."
[added 16th September 2016]
G. Cotti, "Coalescence phenomenon of quantum cohomology of Grassmannians and the distribution of prime numbers" (preprint 08/2016)
[abstract:] "The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of Small Quantum Cohomology of complex Grassmannians is studied. It is shown that surprisingly this frequency is strictly subordinate and highly influenced by the distribution of prime numbers. Two equivalent formulations of the Riemann Hypothesis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associated to the sequence counting non-coalescing Grassmannians, the latter as asymptotic estimate (whose error term cannot be improved) for their distribution function."
[added 16th September 2016]
V. K. Varma, S. Pilati and V. E. Kravtsov, "Conduction in quasi-periodic and quasi-random lattices: Fibonacci, Riemann, and Anderson models" (preprint 07/2016)
[abstract:] "We study the ground state conduction properties of noninteracting electrons in aperiodic but non-random one-dimensional models with chiral symmetry, and make comparisons against Anderson models with non-deterministic disorder. The first model we consider is the Fibonacci lattice, which is a paradigmatic model of quasicrystals; the second is the Riemann lattice, which we define inspired by Dyson's proposal on the possible connection between the Riemann hypothesis and a suitably defined quasicrystal. Our analysis is based on Kohn's many-particle localization tensor defined within the modern theory of the insulating state. In the Fibonacci quasicrystal, where all single-particle eigenstates are critical (i.e., intermediate between ergodic and localized), the noninteracting electron gas is found to be a conductor at most electron densities, including the half-filled case; however, at various specific fillings $\rho$, including the values $\rho = 1/\rho^n$, where $g$ is the golden ratio and $n$ is any integer, the gas turns into an insulator due to spectral gaps. Metallic behaviour is found at half-filling in the Riemann lattice as well; however, in contrast to the Fibonacci quasicrystal, the Riemann lattice is generically an insulator due to single-particle eigenstate localization, likely at all other fillings. Its behaviour turns out to be alike that of the off-diagonal Anderson model, albeit with different system-size scaling of the band-centre anomalies. The advantages of analysing the Kohn's localization tensor instead of other measures of localization familiar from the theory of Anderson insulators (such as the participation ratio or the Lyapunov exponent) are highlighted."
[added 16th September 2016]
R. Ramanathan, M. Túlio Quintino, A. Belén Sainz, G. Murta, R. Augusiak, "On the tightness of correlation inequalities with no quantum violation" (preprint 07/2016)
[abstract:] "We study the faces of the set of quantum correlations, i.e., the Bell and noncontextuality inequalities without any quantum violation. First, we investigate the question whether every proper (tight) Bell inequality for two parties, other than the trivial ones from positivity, normalization and no-signaling can be violated by quantum correlations, i.e., whether the classical Bell polytope or the smaller correlation polytope share any facets with their respective quantum sets. To do this, we develop a recently derived bound on the quantum value of linear games based on the norms of game matrices to give a simple sufficient condition to identify linear games with no quantum advantage. Additionally we show how this bound can be extended to the general class of unique games, illustrating it for the case of three outcomes. We then show as a main result that the paradigmatic examples of correlation Bell inequalities with no quantum violation, namely the non-local computation games do not constitute tight Bell inequalities, not even for the correlation polytope. **We also extend this to an arbitrary prime number of outcomes for a specific class of these games**. We then study the faces in the simplest CHSH Bell scenario of binary dichotomic measurements, and identify edges in the set of quantum correlations in this scenario.
Finally, we relate the non-contextual polytope of single-party correlation inequalities with the cut polytope $CUT(\nabla G)$, where $G$ denotes the compatibility graph of observables in the contextuality scenario and $\nabla G$ denotes the suspension graph of $G$. We observe that there exist tight non-contextuality inequalities with no quantum violation, and furthermore that this set of inequalities is beyond those implied by the Consistent Exclusivity principle."
[added 16th September 2016]
C.M. Bender, D.C. Brody and M.P. Müller, "Hamiltonian for the zeros of the Riemann zeta function" (preprint 08/2016)
[abstract:] "A Hamiltonian operator $\^H$ is constructed with the property that if the eigenfunctions are formally required to obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical limit of $\^H$ is $2xp$, which is consistent with the Berry–Keating conjecture. While $\^H$ is not Hermitian, $i\^H$ is symmetric under $\mathcal{PT}$ reflection. The $\mathcal{PT}$ symmetry appears to be maximally broken, which suggests that the eigenvalues of $\^H$ are real and that the Riemann hypothesis is valid. Further heuristic analysis of $\^H$ supports the conclusion that its eigenvalues are real."
[added 16th September 2016]
F. Bornemann, P.J. Forrester and A. Mays, "Finite size effects for spacing distributions in random matrix theory: Circular ensembles and Riemann zeros" (preprint 08/2016)
[abstract:] "According to Dyson's three fold way, from the viewpoint of global time reversal symmetry there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular orthogonal, unitary and symplectic ensembles, denoted COE, CUE and CSE respectively. For each of these three ensembles and their thinned versions, whereby each eigenvalue is deleted independently with probability $1 - \xi$, we take up the problem of calculating the first two terms in the scaled large $N$ expansion of the spacing distributions. It is well known that the leading term admits a characterisation in terms of both Fredholm determinants and Painlevé transcendents. We show that modifications of these characterisations also remain valid for the next to leading term, and that they provide schemes for high precision numerical computations. In the case of the CUE there is an application to the analysis of Odlyzko's data set for the Riemann zeros, and in that case some further statistics are similarly analysed."
[added 16th September 2016]
K.V. Shajesh, I. Brevik, I. Cavero-Peláez and P. Parashar, "Self-similar plates: Casimir energies" (preprint 07/2016)
[abstract:] "We construct various self-similar configurations using parallel $\delta$-function plates and show that it is possible to evaluate the Casimir interaction energy of these configurations using the idea of self-similarity alone. We restrict our analysis to interactions mediated by a scalar field, but the extension to electromagnetic field is immediate. Our work unveils an easy and powerful method that can be easily employed to calculate the Casimir energies of a class of self-similar configurations. As a highlight, in an example, we determine the Casimir interaction energy of a stack of parallel plates constructed by positioning $\delta$-function plates at the points constituting the Cantor set, a prototype of a fractal. This, to our knowledge, is the first time that the Casimir energy of a fractal configuration has been reported. Remarkably, the Casimir energy of some of the configurations we consider turn out to be positive, and a few even have zero Casimir energy. For the case of positive Casimir energy that is monotonically decreasing as the stacking parameter increases the interpretation is that the pressure of vacuum tends to inflate the infinite stack of plates. We further support our results, derived using the idea of self-similarity alone, by rederiving them using the Green's function formalism. These expositions gives us insight into the **connections between the regularization methods used in quantum field theories and regularized sums of divergent series in number theory**."
[added 16th September 2016]
I. Gálvez-Carrillo, R.M. Kaufmann and A. Tonks, "Three Hopf algebras and their common simplicial and categorical background" (preprint 07/2016)
[abstract:] "We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebras of Goncharov for multiple zeta values, that of Connes–Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common framework."
[added 16th September 2016]
T. Olupitan, C. Lei and A. Vourdas, "An analytic function approach to weak mutually unbiased bases" (preprint 07/2016)
"Quantum systems with variables in $\mathbb{Z}(d)$ are considered, and three different structures are studied. The first is weak mutually unbiased bases, for which the absolute value of the overlap of any two vectors in two different bases is $1/\sqrt{k}$ (where $k\vert d$) or $0$. The second is maximal lines through the origin in the $\mathbb{Z}(d)\times \mathbb{Z}(d)$ phase space. The third is an analytic representation in the complex plane based on Theta functions, and their zeros. It is shown that there is a correspondence (triality) that links strongly these three apparently different structures. For simplicity, the case where $d = p_1\times p_2$, **where $p_1$, $p_2$ are odd prime numbers different from each other, is considered**."
[added 16th September 2016]
D. Broadhurst, "Feynman integrals, $L$-series and Kloosterman moments" (preprint 02/2016)
[abstract:] "This work lies at an intersection of three subjects: quantum field theory, algebraic geometry and number theory, in a situation where dialogue between practitioners has revealed rich structure. It contains a theorem and 7 conjectures, tested deeply by 3 optimized algorithms, on relations between Feynman integrals and $L$-series defined by products, over the primes, of data determined by moments of Kloosterman sums in finite fields. There is an extended introduction, for readers who may not be familiar with all three of these subjects. Notable new results include conjectural evaluations of non-critical $L$-series of modular forms of weights 3, 4 and 6, by determinants of Feynman integrals, an evaluation for the weight 5 problem, at a critical integer, and formulas for determinants of arbitrary size, tested up to 30 loops. It is shown that the functional equation for Kloosterman moments determines much but not all of the structure of the $L$-series. In particular, for problems with odd numbers of Bessel functions, it misses a crucial feature captured in this work by novel and intensively tested conjectures. For the 9-Bessel problem, these lead to an astounding compression of data at the primes."
[added 16th September 2016]
O. Barrett, P. Burkhardt, J. DeWitt, R. Dorward and S.J. Miller, "One-level density for holomorphic cusp forms of arbitrary level" (preprint 04/2016)
"In 2000 Iwaniec, Luo, and Sarnak proved for certain families of $L$-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. We remove the square-free restriction by obtaining a trace formula for arbitrary level by using a basis developed by Blomer and Milicevic, which is of use for other problems as well."
[added 16th September 2016]
V. Blomer, J. Bourgain and Z. Rudnick, "Small gaps in the spectrum of the rectangular billiard" (preprint 04/2016)
"We study the size of the minimal gap between the first $N$ eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio $\alpha$, in comparison to the corresponding quantity for a Poissonian sequence. If $\alpha$ is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size $1/N$, which is essentially consistent with Poisson statistics. We also give related results for a set of $\alpha$'s of full measure. However, on a fine scale we show that Poisson statistics is violated for all $\alpha$. The proofs use a variety of ideas of an arithmetical nature, involving Diophantine approximation, the theory of continued fractions, and **results in analytic number theory. One of our results is conditional on the Riemann Hypothesis.**"
[added 16th September 2016]
F. Qiu, "A necessary condition for the existence of the nontrivial zeros of the Riemann zeta function" (preprint 01/2008)
[abstract:] "Starting from the symmetrical reflection functional equation of the zeta function, we have found that the $\sigma$ values satisfying $\zeta(s) = 0$ must also satisfy $|\zeta(s)| = |\zeta(1 - s)|$. The $\sigma$ values satisfying this requirement are a necessary condition for the existence of the nontrivial zeros. We have shown that $\sigma = 1/2$ is the only numeric solution that satisfies the requirement."
[added 16th September 2016]
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