On Robin’s inequality

Limits and inequalities associated with the Euler–Mascheroni constant

To prove: that the generalized Riemann hypothesis is true, namely, that all the zeta zeros of L -functions are balanced on the one-half number line, using the scale of the harmonic zeta function. Now that the Riemann hypothesis has been proven (in “Summa Characteristica and the Riemann Hypothesis: Scaling Riemann’s Mountain”, Journal of Interdisciplinary Mathematics, Vol. 11 (6) (December 2008)), it can be extended by analytical continuation to prove the generalized hypothesis concerning L-functions. The initial “gene” or generating (one half) principle that proves the Riemann hypothesis also proves that no L-series has a zero with real part on the complex plane other than one half. As a result of this proof, the Birch/Swinnerton-Dyer conjecture is also resolved. Finally, what impact, if any, do these proofs have on security codes based upon prime numbers?

Question – How does the Riemann Hypothesis Support Topological Propulsion and Faster-than-light Travel? Answer – 
 
 a) Using the axiom that there indeed are infinitely many nontrivial zeros on the critical line (calculations have confirmed the hypothesis to be true to over 13 trillion places), the critical line is identified as the y-axis of Wick rotation. This suggests the y-axis is literally infinite and that infinity equals zero. In this case, it is zero distance in time and space. Travelling zero distance is done instantly and is therefore faster-than-light travel. 
 b) Wick rotation is essential to this article’s description of a topological (mathematical) universe and the Riemann hypothesis’ identification with Wick means the hypothesis doesn’t just apply to the distribution of prime numbers but also applies to the fundamental structure of the mathematical universe’s space-time. 
 
 An absolutely essential ingredient of time travel by applied maths is being able to journey into the remote past and future. For this reason, a paragraph about the coexistence of past, present, and future (without such coexistence, time travel could never be a possibility) is included in a section titled “Riemann Hypothesis and Wick Rotation Support Time Travel into the Past and Future”.
 This article provides insights into travelling at significant fractions of – as well as faster than - light, the Higgs boson and field, electroweak interaction, dark matter, dark energy, other dimensions, space-time (eg the expanding-universe question and time travel), quantum mechanics, quantum computers, the Riemann hypothesis, and Unidentified Flying Objects. 
 The article is also a brief summary discussing how the Matrix, the Riemann hypothesis, quaternions, Wick rotation, and imaginary numbers unite topology’s Mobius strip with known and unknown dimensions as well as the Higgs boson, Higgs field, dark matter, dark energy, antigravitons, and the static universe. The article also explains Unidentified Flying Objects as the result of future human technology.

The Riemann hypothesis (RH) states that the non-trivial zeros of the Riemann zeta-function are of the form sn = 1/2+iλn. An improvement of our previous construction to prove the RH is presented by implementing the Hilbert–Polya proposal and furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu–Sprung potential (that capture the average level density of zeros) by recurring to a weighted superposition of Weierstrass functions ∑p W(x, p, D) and where the summation has to be performed over all primes p in order to recapture the connection between the distribution of zeta zeros and prime numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schroedinger equation whose fluctuating potential (relative to the smooth Wu–Sprung potential) has the same functional form as the fluctuating part of the level density of zeros. The second approach to prove the RH relies on the existence of a continuous family of scaling-like operators involving the Gauss–Jacobi theta series. An explicit completion relation ("trace formula") related to a superposition of eigenfunctions of these scaling-like operators is defined. If the completion relation is satisfied, this could be another test of the Riemann Hypothesis. In an appendix, we briefly describe our recent findings showing why the Riemann Hypothesis is a consequence of [Formula: see text]-invariant Quantum Mechanics, because [Formula: see text] where s are the complex eigenvalues of the scaling-like operators. We show why [Formula: see text] invariance requires that s(1 - s) = real , which implies that s is real and/or it lies in the critical Riemann line.

The Riemann Hypothesis states that the Riemann zeta function ζ(z) admits a set of ‘non-trivial’ zeros that are complex numbers supposed to have real part 1/2. Their distribution on the complex plane is thought to be the key to determine the number of prime numbers before a given number. Hilbert and Pólya suggested that the Riemann Hypothesis could be solved through the mathematical tools of physics, finding a suitable Hermitian or unitary operator that describe classical or quantum systems, whose eigenvalues distribute like the zeros of ζ(z). A different approach is that of finding a correspondence between the distribution of the ζ(z) zeros and the poles of the scattering matrix S of a physical system. Our contribution is articulated in two parts: in the first we apply the infinite-components Majorana equation in a Rindler spacetime and compare the results with those obtained with a Dirac particle following the Hilbert-Pólya approach showing that the Majorana solution has a behavior similar to that of massless Dirac particles and finding a relationship between the zeros of zeta end the energy states. Then, we focus on the S-matrix approach describing the bosonic open string scattering for tachyonic states with the Majorana equation. Here we find that, thanks to the relationship between the angular momentum and energy/mass eigenvalues of the Majorana solution, one can explain the still unclear point for which the poles and zeros of the S-matrix of an ideal system that can satisfy the Riemann Hypothesis, exist always in pairs and are related via complex conjugation. As claimed in the literature, if this occurs and the claim is correct, then the Riemann Hypothesis could be in principle satisfied, tracing a route to a proof.

The Euler-Mascheroni constant is calculated by three novel representations\nover these sets respectively: 1) Tur\\'an moments, 2) coefficients of Jensen\npolynomials for the Taylor series of the Riemann Xi function at s=1/2+i.t and\n3) even coefficients of the Riemann Xi function around s=1/2. These findings\nsupport the acceptance of the property of hyperbolicity of Jensen polynomials\nwithin the scope of the Riemann Hypothesis due to exactness on the\napproximations calculated not only for the Euler-Mascheroni constant but also\nfor the Bernoulli numbers and the even derivatives of the Riemann Xi function\nat s=1/2. The new formulas are linked to similar patterns observed in the\nformulation of the Akiyama-Tanigawa algorithm based on the Gregory coefficients\nof second order and lead to understanding the Riemann zeta function as a bridge\nbetween the Gregory coefficients and other relevant sets\n

For \(\mathbb {K}=\mathbb {R}\) or \(\mathbb {C}\), the Hardy–Littlewood inequality for m-linear forms asserts that for \(4\le 2m\le p\le \infty \) there exists a constant \(C_{m,p}^{\mathbb {K}}\ge 1\) such that, for all m-linear forms \(T:\ell _{p}^{n}\times \cdots \times \ell _{p}^{n}\rightarrow \mathbb {K}\), and all positive integers n, Open image in new window This result was proved by Hardy and Littlewood (QJ Math 5:241–254, 1934) for bilinear forms and extended to m-linear forms by Praciano-Pereira (J Math Anal Appl 81:561–568, 1981). The case \(p=\infty \) recovers the Bohnenblust–Hille inequality (Ann Math 32:600–622, 1931). In this paper, among other results, we show that for \(p>2m(m-1)^2\) the optimal constants satisfying the Hardy–Littlewood inequality for m-linear forms are dominated by the best known constants of the corresponding Bohnenblust–Hille inequality. For instance, we show that if \(p>2m(m-1)^2\), then $$\begin{aligned} \textstyle C_{m,p}^{\mathbb {C}}\le \prod \limits _{j=2}^{m}\Gamma \left( 2-\frac{1}{j}\right) ^{\frac{j}{2-2j}}<m^{\frac{1-\gamma }{2}}, \end{aligned}$$ where \(\gamma \) is the Euler–Mascheroni constant.

The motion of a disk in a Langmuir film bounding a liquid substrate is a classical hydrodynamic problem, dating back to Saffman (J. Fluid Mech., vol. 73, 1976, p. 593) who focused upon the singular problem of translation at large Boussinesq number, ${\textit {Bq}}\gg 1$ . A semianalytic solution of the dual integral equations governing the flow at arbitrary ${\textit {Bq}}$ was devised by Hughes et al. (J. Fluid Mech., vol. 110, 1981, p. 349). When degenerated to the inviscid-film limit ${\textit {Bq}}\to 0$ , it produces the value $8$ for the dimensionless translational drag, which is $50\,\%$ larger than the classical $16/3$ -value corresponding to a free surface. While that enhancement has been attributed to surface incompressibility, the mathematical reasoning underlying the anomaly has never been fully elucidated. Here we address the inviscid limit ${\textit {Bq}}\to 0$ from the outset, revealing a singular mechanism where half of the drag is contributed by the surface pressure. We proceed beyond that limit, considering a nearly inviscid film. A naïve attempt to calculate the drag correction using the reciprocal theorem fails due to an edge singularity of the leading-order flow. We identify the formation of a boundary layer about the edge of the disk, where the flow is primarily in the azimuthal direction with surface and substrate stresses being asymptotically comparable. Utilising the reciprocal theorem in a fluid domain tailored to the asymptotic topology of the problem produces the drag correction $(8\,{\textit {Bq}}/{\rm \pi} ) [ \ln (2/{\textit {Bq}}) + \gamma _E+1]$ , $\gamma _E$ being the Euler–Mascheroni constant.

Ramanujan’s formula for the harmonic number

This study focuses on an in-depth investigation of the Riemann zeta function. For this purpose, infinite numbers and rotational infinite numbers, which have been introduced in previous studies published by the author, are used. These numbers are a powerful tool for solving problems involving infinity that are otherwise difficult to solve. Infinite numbers are a superset of complex numbers and can be either complex numbers or some quantification of infinity. The Riemann zeta function can be written as a sum of three rotational infinite numbers, each of which represents infinity. Using these infinite numbers and their properties, a correlation of the non-trivial zeros of the Riemann zeta function with each other is revealed and proven. In addition, an interesting relation between the Euler–Mascheroni constant (γ) and the non-trivial zeros of the Riemann zeta function is proven. Based on this analysis, complex series limits are calculated and important conclusions about the Riemann zeta function are drawn. It turns out that when we have non-trivial zeros of the Riemann zeta function, the corresponding Dirichlet series increases linearly, in contrast to the other cases where this series also includes a fluctuating term. The above theoretical results are fully verified using numerical computations. Furthermore, a new numerical method is presented for calculating the non-trivial zeros of the Riemann zeta function, which lie on the critical line. In summary, by using infinite numbers, aspects of the Riemann zeta function are explored and revealed from a different perspective; additionally, interesting mathematical relationships that are difficult or impossible to solve with other methods are easily analyzed and solved.

Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: Hn∼γ+logn−∑k=1∞Bkk·nk, where Bk is the Bernoulli numbers. In this paper, we rewrite Ramanujan’s harmonic number expansion into a similar form of Euler’s asymptotic expansion as n approaches infinity: Hn∼γ+c0(h)log(q+h)−∑k=1∞ck(h)k·(q+h)k, where q=n(n+1) is the nth pronic number, twice the nth triangular number, γ is the Euler–Mascheroni constant, and ck(x)=∑j=0kkjcjxk−j, with ck is the negative of the median Bernoulli numbers. Then, 2cn=∑k=0nnkBn+k, where Bn is the Bernoulli number. By using the result obtained, we present two general Ramanujan’s asymptotic expansions for the nth harmonic number. For example, Hn∼γ+12log(q+13)−1180(q+13)2∑j=0∞bj(r)(q+13)j1/r as n approaches infinity, where bj(r) can be determined.

We present some properties associated with the monotonicity and the complete monotonicity of the psi (or digamma) function and establish the higher-order estimate for the familiar Euler–Mascheroni constant γ. Relevant connections of the results presented here with those derived in earlier works are also pointed out.

The aim of this paper is to propose a new sequence that approximates the Euler–Mascheroni constant which converges faster towards its limit and to establish new inequalities for this constant.

Estimations of psi function and harmonic numbers

In this work, new types of finite subgroups are given, they are called prime gaps additive subgroups (S, +p) of Zp and prime gaps multiplication subgroups (S, p) of Zp*. Also, new practical algorithm to track prime factors for any composite numbers is introduced. This new method depends on the general rule for prime numbers that has one variable (n). The new rule is very important because it exactly gives the number of primes for any interval of integer numbers. Also, the large gaps between n-th primes are given in this paper. Next, this formula is used to make application by Excel in order to exhibit the results for any integer number if it is prime or not at polynomial time. By this new method, we can find straightforwardly more than (97.5%) of the composite numbers while the rest of the percentage needs polynomial-time. In addition, we can use this rule to exhibit all prime numbers between the smallest prim number and any number x in sequence list. Finally, by this work the following points are considered; The distribution for any sequence of prime numbers is given, Our results better than Riemann Hypothesis (see Tables 3 and 4), For any large integer we can find its factorization in polynomial time by a classical computer.