The UV prolate spectrum matches the zeros of zeta

In this paper we introduce the real valued real analytic function kappa(t) implicitly defined by exp(2 pi i kappa(t)) = -exp(-2 i theta(t)) * (zeta'(1/2-it)/zeta'(1/2+it)) and kappa(0)=-1/2. (where theta(t) is the function appearing in the known formula zeta(1/2+it)= Z(t) * e^{-i theta(t)}). By studying the equation kappa(t) = n (without making any unproved hypotheses), we will show that (and how) this function is closely related to the (exact) position of the zeros of Riemann's zeta(s) and zeta'(s). Assuming the Riemann hypothesis and the simplicity of the zeros of zeta(s), it will follow that the ordinate of the zero 1/2 + i gamma_n of zeta(s) will be the unique solution to the equation kappa(t) = n.

Requiring monotonicity of the product of the exponential with a function, we prove an inequality for the Mellin transform of the function. For a certain class of functions, the log-convex property is proved. Applications of the result in proving inequalities and the log-convex property for the extended zeta function, the Riemann zeta function, and the Macdonald function are discussed.

Contour integral representations of Riemann's Zeta function and Dirichlet's Eta (alternating Zeta) function are presented and investigated. These representations flow naturally from methods developed in the 1800s, but somehow they do not appear in the standard reference summaries, textbooks, or literature. Using these representations as a basis, alternate derivations of known series and integral representations for the Zeta and Eta function are obtained on a unified basis that differs from the textbook approach, and results are developed that appear to be new.

The history of problems of evaluation of series associated with the Riemann Zeta function can be traced back to Christian Goldbach (1690–1764) and Leonhard Euler (1707–1783). Many di¤erent techniques to evaluate various series involving the Zeta and related functions have since then been developed. The authors show how elegantly certain families of series involving the Zeta function can be evaluated by starting with a single known identity for the generalized (or Hurwitz) Zeta function. Some special cases and their connections with already developed series involving the Zeta and related functions are also considered.

Prime numbers are the building blocks of our arithmetic; however, their distribution still poses fundamental questions. Riemann showed that the distribution of primes could be given explicitly if one knew the distribution of the nontrivial zeros of the Riemann zeta(s) function. According to the Hilbert-Pólya conjecture, there exists a Hermitian operator of which the eigenvalues coincide with the real parts of the nontrivial zeros of zeta(s) . This idea has encouraged physicists to examine the properties of such possible operators, and they have found interesting connections between the distribution of zeros and the distribution of energy eigenvalues of quantum systems. We apply the Marchenko approach to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the zeta(s) function. We demonstrate the multifractal nature of these potentials by measuring the Rényi dimension of their graphs. Our results offer hope for further analytical progress.

We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions providing analytic continuation throughout the whole complex plane. Additionally, we demonstrate some series representations for the initial Stieltjes constants appearing in the Laurent expansion of the Hurwitz zeta function. A particular point of elaboration in these developments is the hypergeometric form and its equivalents for certain derivatives of the incomplete Gamma function. Finally, we evaluate certain integrals including ∫ \tiny {Re} s = c ζ ( s ) s d s \int _{\mbox {\tiny {Re}} s=c} {{\zeta (s)} \over s} ds and ∫ \tiny {Re} s = c η ( s ) s d s \int _{\mbox {\tiny {Re}} s=c} {{\eta (s)} \over s} ds , with ζ \zeta the Riemann zeta function and η \eta its alternating form.

Fors∈ℂ, the Euler zeta function and the Hurwitz-type Euler zeta function are defined byζE(s)=2∑n=1∞((−1)n/ns), andζE(s,x)=2∑n=0∞((−1)n/(n+x)s). Thus, we note that the Euler zeta functions are entire functions in whole complexs-plane, and these zeta functions have the values of the Euler numbers or the Euler polynomials at negative integers. That is,ζE(−k)=Ek∗, andζE(−k,x)=Ek∗(x). We give some interesting identities between the Euler numbers and the zeta functions. Finally, we will give the new values of the Euler zeta function at positive even integers.

To prove: Riemann’s hypothesis that all zeta zeros have real part one half. Premise one : the (characteristic) harmonic triangle of Pascal is characterized by the harmonic series. Premise two: when s equals one in the zeta function, the function is identical with the harmonic series. Three : when s equals any complex number, the zeta function is the same as the complex number plane. Four: since the harmonic pattern of the series in the triangle of Pascal completely carries over into the zeta function, and the zeta function charts the complete complex plane (premises one through three combined), whatever places restrictions on the Pascal harmonic pattern also places constraints on the number pattern of the complex plane. Five : the defining (characteristic) restriction responsible for the generation of the whole triangle-plane is the number of both the binomial theorem and the geometrical expansion, which is one half. Because this fraction-interval is recursively present throughout the entire harmonic series, it ties all of the zeta zeros to the real line one half. By recursion we can follow one half as the real part of the zeta zeros forward throughout the entire harmonic series and onto the complex plane. Conclusion: all zeta zeros have real part one half. Since the premises are all true, the conclusion must also be true. Therefore, the Riemann hypothesis is correct. Nature is the realization of the simplest conceivable mathematical ideas. A physicist’s greatest challenge is to uncover the fundamental and universal laws from which a process of simple deduction can lead to a picture of the world. [.] This is what Leibniz felicitously called a ‘re-established’ harmony. Albert Einstein, 1918

Four basic problems in Riemann’s original paper are found. The Riemann hypothesis becomes meaningless. 1. It is proved that on the real axis of complex plane, the Riemann Zeta function equation holds only at point Re(s)=1/2 (s = a+ib). However, at this point, the Zeta function is infinite, rather than zero. At other points of real axis, the two sides of Zeta function equation are contradictory. When one side is finite, another side may be infinite. 2. An integral item around the original point of coordinate system was neglected when Riemann deduced the integral form of Zeta function. The item was convergent when Re(s) > 1 but divergent when Re(s) 0. At point x = 0, the formula is meaningless. However, the lower limit of Zeta function integral is x = 0, so the formula can not be used. 4. A formula of Jacobi function was used to prove the symmetry of Zeta function equation. The applicable condition of this formula was also x > 0. However, the lower limit of integral in the deduction was x=0. So this formula can not be used too. The zero calculation of Riemann Zeta function is discussed at last. It is pointed out that because approximate methods are used, they are not the real zeros of strict Riemann Zeta function.

The Riemann zeta (ζ) function ζ(s) = ∞ n=1 1 n s is valid for all complex number s = x + iy : Re(s) > 1, for the line x = 1.Euler-Riemann found that the function equals zero for all negative even integers: -2, -4, -6, • • • (commonly known as trivial zeros) has an infinite number of zeros in the critical strip of complex numbers between the lines x = 0 and x = 1.Moreover, it was well known to him that all non-trivial zeros are exhibiting symmetry with respect to the critical line x = 1 2 .As a result, Riemann conjectured that all of the non-trivial zeros are on the critical line, this hypothesis is known as the Riemann hypothesis.The Riemann zeta function plays a momentous part while analyzing the number theory and has applications in applied statistics, probability theory and Physics.The Riemann zeta function is closely related to one of the most challenging unsolved problems in mathematics (the Riemann hypothesis) which has been classified as the 8th of Hilbert's 23 problems.This function is useful in number theory for investigating the anomalous behavior of prime numbers.If this theory is proven to be correct, it means we will be able to know the sequential order of the prime numbers.Numerous approaches have been applied towards the solution of this problem, which includes both numerical and geometrical approaches, also the Taylor series of the Riemann zeta function, and the asymptotic properties of its coefficients.Despite the fact that there are around 10 13 , non-trivial zeros on the critical line, we cannot assume that the Riemann Hypothesis (RH) is necessarily true unless a lucid proof is provided.Indeed, there are differing viewpoints not only on the Riemann Hypothesis's reliability, but also on certain basic conclusions see for example [16] in which the author justifies the location of non-trivial zero subject to the simultaneous occurrence of ζ(s) = ζ(1 -s) = 0, and omitting the impact of an indeterminate form ∞.0, that appears in Riemann's approach.In this study we also consider the simultaneous occurrence ζ(s) = ζ(1 -s) = 0 but we adopt an element-wise approach of the Taylor series by expanding n -x for all n = 1, 2, 3, • • • at the real parts of the non-trivial zeta zeros lying in the critical strip for s = α + iy is a non-trivial zero of ζ(s), we first expand each term n -x at α then at 1 -α.Then In this sequel, we evoke the simultaneous occurrence of the non-trivial zeta function zeros ζ(s) = ζ(1 -s) = 0, on the critical strip by the means of different representations of Zeta function.Consequently, proves that Riemann Hypothesis is likely to be true.

Some classes of infinite series associated with the Riemann Zeta and Polygamma functions and generalized harmonic numbers

Five basic mistakes are found in the Riemann’s original paper proposed in 1859. The Riemann hypothesis becomes meaningless. 1. It is proved that on the real axis of complex plane and in the domain of the function, when the left-hand side of equation is finite, the right-hand side may be infinite, and vice versa. The Riemann Zeta function equation holds only at the point Re(s) = 1/2(s= a+ \(i b\)) . However, at this point, the Zeta function is infinite, rather than zero, to make Riemann hypothesis untenable. 2. An integral item around the original point of coordinate system was ignored when Riemann deduced the integral form of Zeta function. The item was convergent when Re(s) > 1but divergent when Re(s) < 1 .The integral form of Zeta function does not change the divergence of its series form. 3. A summation formula was used in the deduction of the integral form of Zeta function. The applicative condition of this formula is x > 0 . At point x = 0 , the formula is meaningless. However, the lower limit of Zeta function’s integral is x = 0 , so the formula can not be used. 4. Because the integral lower limit of Zeta function is zero, the integrand function is not uniformly convergent, so integral sign and sum sign can not be exchanged. But Riemann made them interchangeable, resulting in that the integral form of Zeta function is untenable. 5.The formula \(\theta(x)=\sqrt{x} \theta(1 / x)\) of Jacobi function was used to prove the symmetry of Zeta function equation. The applicable condition of this formula is also x > 0 . Because the lower limit of integral in the deduction was x = 0 , this formula can not be used too. The zero calculation of Riemann Zeta function is discussed at last. It is pointed out that because approximate methods are used, the analytic property of the original function is destroyed and the Cauchy-Riemann equation can not be satisfied. So they are not the real zeros of strict Riemann Zeta function.

Certain families of series associated with the Hurwitz–Lerch Zeta function

On the one hand, the Fermi–Dirac and Bose–Einstein functions have been extended in such a way that they are closely related to the Riemann and other zeta functions. On the other hand, the Fourier transform representation of the gamma and generalized gamma functions proved useful in deriving various integral formulae for these functions. In this paper, we use the Fourier transform representation of the extended functions to evaluate integrals of products of these functions. In particular, we evaluate some integrals containing the Riemann and Hurwitz zeta functions, which had not been evaluated before.

We initiate the study of spectral zeta functions $\zeta_X$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function $\zeta(s)$ is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of $\zeta(s)$. We relate $\zeta_{\mathbb{Z}}$ to Euler's beta integral and show how to complete it giving the functional equation $\xi_{\mathbb{Z}}(1-s)=\xi_{\mathbb{Z}}(s)$. This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions $d$ we provide a meromorphic continuation of $\zeta_{\mathbb{Z}^d}(s)$ to the whole plane and identify the poles. From our aymptotics several known special values of $\zeta(s)$ are derived as well as its non-vanishing on the line $Re(s)=1$. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function $F_1$ via an Euler-type integral formula due to Picard.