Riemann's Functional Equation Research Articles

Bicomplex numbers as a normal complexified f-algebra

The algebra B of bicomplex numbers is viewed as a complexification of the Archimedean f-algebra of hyperbolic numbers D. This lattice-theoretic approach allows us to establish new properties of the so-called D-norms. In particular, we show that D-norms generate the same topology in B. We develop the D-trigonometric form of a bicomplex number which leads us to a geometric interpretation of the nth roots of a bicomplex number in terms of polyhedral tori. We use the concepts developed, in particular that of Riesz subnorm of a D-norm, to study the uniform convergence of the bicomplex zeta and gamma functions. The main result of this paper is the generalization to the bicomplex case of the Riemann functional equation and Euler's reflection formula.

Functional equation and zeros on the critical line of the quadrilateral zeta function

For 0<a≤1/2, we define the quadrilateral zeta function Q(s,a) using the Hurwitz and periodic zeta functions and show that Q(s,a) satisfies Riemann's functional equation studied by Hamburger, Heck and Knopp. Moreover, we prove that for any 0<a≤1/2, there exist positive constants A(a) and T0(a) such that the number of zeros of the quadrilateral zeta function Q(s,a) on the line segment from 1/2 to 1/2+iT is greater than A(a)T whenever T≥T0(a).

Functional equations for regularized zeta-functions and diffusion processes

We discuss modifications in the integral representation of the Riemann zeta-function that lead to generalizations of the Riemann functional equation that preserves the symmetry s → (1 − s) in the critical strip. By modifying one integral representation of the zeta-function with a cut-off that does exhibit the symmetry x ↦ 1/x, we obtain a generalized functional equation involving Bessel functions of second kind. Next, with another cut-off that does exhibit the same symmetry, we obtain a generalization for the functional equation involving only one Bessel function of second kind. Some connection between one regularized zeta-function and the Laplace transform of the heat kernel for the Euclidean and hyperbolic space is discussed.

Numerical Calculations to Grasp a Mathematical Issue Such as the Riemann Hypothesis

This article presents the use of data processing to apprehend mathematical questions such as the Riemann Hypothesis (RH) by numerical calculation. Calculations are performed alongside graphs of the argument of the complex numbers ζ ( x + i y ) = a + i b and ξ ( x + i y ) = p + i q , in the critical strip. On the one hand, the two-dimensional surface angle tan − 1 ( b / a ) of the Riemann Zeta function ζ is related to the semi-angle of the fractional part of y 2 π ln ( y 2 π e ) and, on the other hand, the Ksi function ξ of the Riemann functional equation is analyzed with respect to the coordinates ( x , 1 − x ; y ) . The computation of the power series expansion of the ξ function with its symmetry analysis highlights the RH by the underlying ratio of Gamma functions inside the ξ formula. The ξ power series beside the angle of both surfaces of the ζ function enables to exhibit a Bézout identity a u + b v ≡ c between the components ( a , b ) of the ζ function, which illustrates the RH. The geometric transformations in complex space of the Zeta and Ksi functions, illustrated graphically, as well as series expansions, calculated by computer, make it possible to elucidate this mathematical problem numerically. A final theoretical outlook gives deeper insights on the functional equation’s mechanisms, by adopting a computer–scientific perspective.

The Euler–Maclaurin–Siegel and Abel–Plana summation formulae for the entire Riemann functional equation to handle the Riemann hypothesis

In this article, with the aid of the entire Riemann functional equation (ERFE), defined by ξs=12ss−1π−s2Γs2ςs, where s is a complex variable, Γs is the Euler’s gamma function, and ςs is the Riemann Zeta function (RZF), the Euler–Maclaurin–Siegel summation formula (EMSSF) and the Abel–Plana summation formula (APSF) are addressed to prove the Riemann hypothesis (RH) for the first time. The theorems for the ERFE are presented, and the complex zeros for the ERFE and RZF are also discussed in detail. The presented results are accurately and efficiently proposed to find the critical line of the ERFE.

A Proof of the Riemann's Hypothesis

We present a proof of the Riemann's Zeta Hypothesis, based on asymptotic expansions and operations on series. We use the symmetry property presented by Riemann's functional equation to extend the proof to the whole set of complex numbers C. The advantage of our method is that it only uses undergraduate math which makes it accessible to a wider audience.

A formula for the Hurwitz zeta function, Riemann's functional equation and certain integral representations

There is a well-known formula for the Hurwitz zeta function which implies the functional equation for the Riemann zeta function. We give a new proof of that formula and recover certain integral representations for the Hurwitz and Riemann zeta functions.

Jacob’s ladders, factorization, and metamorphoses as an appendix to the Riemann functional equation for ζ(s) on the critical line

Jacob’s ladders, factorization, and metamorphoses as an appendix to the Riemann functional equation for ζ(s) on the critical line

Further exploration of Riemann's functional equation

A previous exploration of the Riemann functional equation that focussed on the critical line, is extended over the complex plane. Significant results include a simpler derivation of the fundamental equation developed previously, and its generalization from the critical line to the complex plane. A simpler statement of the relationship that exists between the real and imaginary components of $\zeta(s)$ and $\zeta^{\prime}(s)$ on opposing sides of the critical line is developed, reducing to a simpler statement of the same result on the critical line. An analytic expression is obtained for the sum of the arguments of $\zeta(s)$ on opposite sides of the critical line, reducing to the analytic expression for $arg(\zeta(1/2+i\rho))$ first obtained in the previous work. Relationships are obtained between various combinations of $|\zeta(s)|$ and $|\zeta^{\prime}(s)|$, particularly on the critical line, and it is demonstrated that $arg(\zeta(1/2+i\rho))$ and $arg(\zeta^{\prime}(1/2+i\rho))$ uniquely define $|\zeta(1/2+i\rho)|$. A comment is made about the utility of such results as they might apply to putative proofs of Riemann's Hypothesis (RH).

Further exploration of Riemann's functional equation

A previous exploration of the Riemann functional equation that focussed on the critical line, is extended over the complex plane. Significant results include a simpler derivation of the fundamental equation developed previously, and its generalization from the critical line to the complex plane. A simpler statement of the relationship that exists between the real and imaginary components of $\zeta(s)$ and $\zeta^{\prime}(s)$ on opposing sides of the critical line is developed, reducing to a simpler statement of the same result on the critical line. An analytic expression is obtained for the sum of the arguments of $\zeta(s)$ on opposite sides of the critical line, reducing to the analytic expression for $arg(\zeta(1/2+i\rho))$ first obtained in the previous work. Relationships are obtained between various combinations of $|\zeta(s)|$ and $|\zeta^{\prime}(s)|$, particularly on the critical line, and it is demonstrated that $arg(\zeta(1/2+i\rho))$ and $arg(\zeta^{\prime}(1/2+i\rho))$ uniquely define $|\zeta(1/2+i\rho)|$. A comment is made about the utility of such results as they might apply to putative proofs of Riemann's Hypothesis (RH).

Jacob’s ladders, factorization, and metamorphoses as an appendix to the Riemann functional equation for $\zeta(s)$ on the critical line

Jacob’s ladders, factorization, and metamorphoses as an appendix to the Riemann functional equation for $\zeta(s)$ on the critical line

A Note on the $S_2(\delta)$ Distribution and the Riemann Xi Function

The theory of $S_2(\delta)$ family of probability distributions is used to give a derivation of the functional equation of the Riemann xi function. The $\delta$ deformation of the xi function is formulated in terms of the $S_2(\delta)$ distribution and shown to satisfy Riemann's functional equation. Criteria for simplicity of roots of the xi function and for its simple roots to satisfy the Riemann hypothesis are formulated in terms of a differentiability property of the $S_2(\delta)$ family. For application, the values of the Riemann zeta function at the integers and of the Riemann xi function in the complex plane are represented as integrals involving the Laplace transform of $S_2.$

The Fourier inversion and the Riemann functional equation

We prove that the Riemann functional equation can be recovered by the Mellin transforms of essentially all the absolutely integrable functions. The present analysis shows also that the Riemann functional equation is equivalent to the Fourier inversion formula. We introduce the notion of a λ -pair of absolutely integrable functions and show that the components of the λ -pair satisfy an identity involving convolution type products.

Proof of Bernhard Riemann’s Functional Equation using Gamma Function

This study shows the use of gamma function to prove the Riemann functional equation. Two approaches had been used to solve this problem: first the value of t in the definition of the gamma function had been changed to pi nu x if only if sigma is greater than zero in the complex plane. Secondly, the Poisson summation formula is used to show that zeta has a simple pole at s = 1 with residue 1, we had found that Riemann zeta function depended intimately on properties of gamma function, which was a new gate for solving complex problems related to zeta function.

Easy proofs of Riemann’s functional equation for $\zeta (s)$ and of Lipschitz summation

We present a new, simple proof, based upon Poisson summation, of the Lipschitz summation formula. A conceptually easy corollary is the functional relation for the Hurwitz zeta function. As a direct consequence we obtain a short, motivated proof of Riemann's functional equation for ((s). INTRODUCTION We present a short and motivated proof of Riemann's functional equation for Riemann's zeta function ((s) = EZ l ' initially defined in the half plane Re(s) > 1. In fact we prove the slightly more general functional relation for the Hurwitz zeta function

On Solutions to Riemann's Functional Equation

In 1921 Hamburger proved that Riemann's functional equation characterizes the Riemann zeta function in the space of functions representable by ordinary Dirichlet series satisfying certain regularity conditions. We consider solutions to a more general functional equation with real weight k. In the case of Hamburger's theorem, k = $$ - \frac{1}{2}$$ . We show that, under suitable conditions, the generalized functional equation admits no nontrivial solutions for k > 0 unless k = $$ - \frac{1}{2}$$ . Our proof generalizes an elegant proof of Hamburger's theorem given by Siegel, and employs a generalized integral transform.

On Dirichlet series satisfying Riemann's functional equation

According to convention, Hamburger's theorem (1921) says-roughly-that Riemann's ζ(s) is uniquely determined by its functional equation. In 1944 Hecke pointed out that there are two distinct versions of Hamburger's theorem. Hecke's remark has led me, in examining just how “rough” the convention is, to prove that, with a weakening of certain auxiliary conditions, there are infinitely many linearly independent solutions of Riemann's functional equation (Theorem 1). In Theorem 1, as in Hamburger's theorem, the “weight” parameter is 1/2. In Theorem 2 we obtain stronger results when this parameter is ≧2: a “Mittag-Leffler” theorem for Dirichlet series with functional equations.

An infinite family of zeta functions indexed by Hermite polynomials

Riemann's hypothesis has challenged many mathematicians and, yet, still remains an open problem. This work deals with the construction of an infinite family of zeta functions, denoted ζ 2 q, 0 ( s) for q = 0, 1, 2 …, which (1) satisfy Riemann's functional equation, and (2) generalize many properties of the Riemann zeta function, ζ( s). In Section 3 we show that ζ 2 q, 0 ( s) = (2 q )! ζ ( s) P q ( s), where P q ( s) is a polynomial in the variable s of degree q . It is true that for small q , the zeroes of P q ( s)have real part equal to one-half, are simple, and are distinct. We conjecture that for all q = 1, 2,…, the zeroes of P q ( s) have real part equal to one-half, are simple, and are distinct, and we hope to prove this conjecture in a paper which is to follow. It is important to note that before one can establish the properties of ζ 2 q, 0 ( s) one must first generalize the Jacobi Inversion Formula. This generalization can be found in Section 2.

Riemann's functional equation

Riemann's functional equation

On Riemann's Functional Equation with Multiple Gamma Factors

(1.3) (2P7r)-isAl(s)cV(s) = (2P7Tr)4 ( S)M( S) of the following description. We have (1.4) A1(s) = F1' (p fs + Cf), (1.5) A2(S) = ll'+1 F(pS + Ce), where G > 1, H-G > 1, HG ;G. The numbers c1, * C , cH are unrestricted complex constants. The constants p1, ** , PH however are real numbers, and (1.6) PA > ?, h 1, * H (1.7) Ah + Pa + * + PHl= Actually we will assume more precisely that we have (1.8) PI+ ** + Pa = 1/2 =PG+l + * * * + PH but only after verifying first that the possibility of (1.9) PA + *-+ Pa # Pail + *-+ PH is not meaningful to our contexts. Finally, we will always assume that the positive constant P in (1.3) has a specified value, namely (1. 10) P = P1P' P2 * ... *,PhP. If P were left unspecified it would burden the wording of our theorems unprofitably. As in [3] and [1], a solution {'v(s), 0b(s)} of (1.3) consists of the following data. Firstly there are given two Dirichlet series with unrelated ex29