A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function

The completed zeta function $\xi(s)$ is expanded in MacLaurin series (infinite polynomial), which can be further expressed as infinite product (Hadamard product) of quadratic factors by its complex conjugate zeros $\rho_i=\alpha_i +j\beta_i, \bar{\rho}_i=\alpha_i-j\beta_i, 0<\alpha_i<1, \beta_i\neq 0, i\in \mathbb{N}$ are natural numbers from 1 to infinity, $\rho_i$ are in order of increasing $|\rho_i|=\sqrt{\alpha_i^2+\beta_i^2}$, i.e., $|\rho_1|<|\rho_2|\leq|\rho_3|\leq |\rho_4|, \cdots$, together with $\beta_1<\beta_2\leq\beta_3\leq\beta_4, \cdots$. Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\xi(0)\prod_{i\in \mathbb{N}}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)} =\xi(0)\prod_{i\in \mathbb{N}}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(1-s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}$$ which, by Lemma 3, is equivalent to $$(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}, \text{from 1 to infinity.}$$ with only valid solution $\alpha_i= \frac{1}{2}$ (another solution $s=\frac{1}{2}$ is invalid due to obvious contradiction). Thus, a proof of the Riemann Hypothesis is achieved.

Riemann Zeta Based Surge Modelling of Continuous Real Functions in Electrical Circuits

Riemann zeta is defined as a function of a complex variable that analytically continues the sum of the Dirichlet series, when the real part is greater than unity. In this paper, the Riemann zeta associated with the finite energy possessed by a 2mm radius, free falling water droplet, crashing into a plane is considered. A modified zeta function is proposed which is incorporated to the spherical coordinates and real analysis has been performed. Through real analytic continuation, the single point of contact of the drop at the instant of touching the plane is analyzed. The zeta function is extracted at the point of destruction of the drop, where it defines a unique real function. A special property is assumed for some continuous functions, where the function’s first derivative and first integral combine together to a nullity at all points. Approximate reverse synthesis of such a function resulted in a special waveform named the dying-surge. Extending the proposed concept to general continuous real functions resulted in the synthesis of the corresponding function’s Dying-surge model. The Riemann zeta function associated with the water droplet can also be modeled as a dying–surge. The Dying- surge model corresponds to an electrical squeezing or compression of a waveform, which was originally defined over infinite arguments, squeezed to a finite number of values for arguments placed very close together with defined final and penultimate values. Synthesized results using simulation software are also presented, along with the analysis. The presence of surges in electrical circuits will correspond to electrical compression of some unknown continuous, real current or voltage function and the method can be used to estimate the original unknown function.

Proof of the functional equation for the Riemann zeta-function

In this article, we shall prove a result which enables us to transfer from finite to infinite Euler products. As an example, we give two new proofs of the infinite product for the sine function depending on certain decompositions. We shall then prove some equivalent expressions for the functional equation, i.e. the partial fraction expansion and the integral expression involving the generating function for Bernoulli numbers. The equivalence of the infinite product for the sine functions and the partial fraction expansion for the hyperbolic cotangent function leads to a new proof of the functional equation for the Riemann zeta function.

Investigation of a Control Function Determining the Location of the Zeros as a Possible Proof of the Riemann Hypothesis.

The article examines the control function in relation to the distribution of Zeros on thecritical line x = 0,5. To confirm this hypothesis, it will be necessary to perform a large number ofstatistical analyzes of the distribution of non-trivial zero points of the Riemann Zeta function.

A Quadruple Integral Involving Chebyshev Polynomials Tn(x): Derivation and Evaluation

The aim of the current document is to evaluate a quadruple integral involving the Chebyshev polynomial of the first kind Tn(x) and derive in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. The zero distribution of almost all Hurwitz-Lerch zeta functions is asymmetrical. All the results in this work are new.

Zeta functions of the 3-dimensional almost-Bieberbach groups

The subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group can be expressed as finite sums of Dirichlet series admitting Euler product factorization. We compute these series except for a finite number of local factors when the group is virtually nilpotent of Hirsch length 3. We deduce that they can be meromorphically continued to the whole complex plane and that they satisfy local functional equations. The complete computation (with no exception of local factors) is presented for those groups that are also torsion-free, that is, for the 3-dimensional almost-Bieberbach groups.

A weighted graph zeta function involved in the Szegedy walk

We define a new weighted zeta function for a finite graph and obtain its determinant expression. This result gives the characteristic polynomial of the transition matrix of the Szegedy walk on a graph.

Harmonic series with polylogarithmic functions

Introduction/purpose: Some sums of the polylogarithmic function associated with harmonic numbers are established. Methods: The approach is based on using the summation methods. Results: This paper generalizes the results of the zeta function series associated with the harmonic numbers. Conclusions: Various interesting series as the consequence of the generalization are obtained.

On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function

We present several formulae for the large t t asymptotics of the Riemann zeta function ζ ( s ) \zeta (s) , s = σ + i t s=\sigma +i t , 0 ≤ σ ≤ 1 0\leq \sigma \leq 1 , t > 0 t>0 , which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum ∑ a b n − s \sum _a^b n^{-s} for certain ranges of a a and b b . In addition, we present precise estimates relating this sum with the sum ∑ c d n s − 1 \sum _c^d n^{s-1} for certain ranges of a , b , c , d a, b, c, d . We also study a two-parameter generalization of the Riemann zeta function which we denote by Φ ( u , v , β ) \Phi (u,v,\beta ) , u ∈ C u\in \mathbb {C} , v ∈ C v\in \mathbb {C} , β ∈ R \beta \in \mathbb {R} . Generalizing the methodology used in the study of ζ ( s ) \zeta (s) , we derive asymptotic formulae for Φ ( u , v , β ) \Phi (u,v, \beta ) .