Papers by Jose Javier Garcia Moreta
viXra, 2011
We present a Berry-Keating model with �periodic� conditions in the dilation group ... (see paper)
An approach to Pi(x) function in number theory
viXra, 2010
ABSTRACT: In this paper we review some results of our previous papers involving Riemann Hypothesi... more ABSTRACT: In this paper we review some results of our previous papers involving Riemann Hypothesis in the sense of Operator theory (Hilbert-Polya approach) and the application of the negative values of the Zeta function (1 s) to the divergent integrals 1 0 s x dx and to the problem of defining a consistent product of distributions of the form ( ) ( ) m n D x D x , in this paper we present new results of how the sums over the non-trivial zeros of the zeta function h( ) can be related to the Mangoldt function 0 (x) assuming Riemann Hypothesis.Throughout the paper we will use the notation ( ) ( ) R s s meaning that we use the zeta regularization for the divergent series 0 s n n s>0 or s=0
In this paper we present a method to get the prime counting function (x) and other arithmetical ... more In this paper we present a method to get the prime counting function (x) and other arithmetical functions than can be generated by a Dirichlet series, first we use the general variational method to derive the solution for a Fredholm Integral equation of first kind with symmetric Kernel K(x,y)=K(y,x), after that we find another integral equations with Kernels K(s,t)=K(t,s) for the Prime counting function and other arithmetical functions generated by Dirichlet series, then we could find a solution for (x) and
In this paper we present a method to deal with divergences in perturbation theory using the metho... more In this paper we present a method to deal with divergences in perturbation theory using the method of the Zeta regularization, first of all we use the Euler-Mc Laurin Sum formula to associate the divergent integral to a divergent sum in the form 1+2^{m}+3^{m}+... After that we find a recurrence formula for the integrals and apply zeta regularization techniques to obtain finite results for the divergent series. (Through all the paper we use the notation m, for the power of the modulus of p, so we must not confuse it with the value of the mass of the quantum particle) changes: added Bibliography about ramanujan Resummation and references of Zeta regularization in Hardy,s Book "divergent series".
In this paper we use the Mellin convolution theorem, which is related to Perron's formula. Al... more In this paper we use the Mellin convolution theorem, which is related to Perron's formula. Also we introduce new explicit formulae for arithmetic function which generalize the explicit formulae of Weil for other arithmetic functions different from the Von-Mangoldt function.
In this paper we give an integral equation satisfied by the gramm series based on the use of the ... more In this paper we give an integral equation satisfied by the gramm series based on the use of the Borel transform In Mathematics the Gramm series is define as the infinite series (1)
A New Approach to the Renormalization of Uv Divergences Using Zeta Regularization Techniques
Using the theory of distributions and Zeta regularization we manage to give a definition of produ... more Using the theory of distributions and Zeta regularization we manage to give a definition of product for Dirac delta distributions, we show how the fact of one can be define a coherent and finite product of dDirac delta distributions is related to the regularization of divergent integrals m s a x dx ∞ − ∫ a >0 and Fourier series, for a Fourier series making a Taylor substraction we can define a regular part ( ) reg F u defined as a function for every 'u' plus a dirac delta series ( ) 0 ( ) N i i i c u δ = ∑ , which is divergent for u=0 , we show then how ( ) (0) i δ can be regularized using a combination of Euler-Mclaurin formula and analytic continuation for the series 0 ( ) k i i k ζ ∞ = = − ∑ PRODUCT OF DIRAC DELTA DISTRIBUTIONS ( ) ( ) m x δ x ( ) ( ) n x δ One of the problems with distributions , as proved by Schartz (see ref [1] ) is that we can not (in general) define a coherent product of distributions, for example ( )
In this paper we discuss a method to express the Prime counting function as a "sum" over Non-triv... more In this paper we discuss a method to express the Prime counting function as a "sum" over Non-trivial zeros of Riemann Zeta function, using techniques from Analytic Number Theory, also we apply our results to the sum over primes of any function Sum(p<x) f(p)
We give an interpretation of the Riemann Xi-function ξ(s) as the quotient of two functional deter... more We give an interpretation of the Riemann Xi-function ξ(s) as the quotient of two functional determinants of an Hermitian Hamiltonian H = H † . To get the potential of this Hamiltonian we use the WKB method to approximate and evaluate the spectral Theta function Θ(t) = n exp(−tγ 2 n ) over the Riemann zeros on the critical strip 0 < Re(s) < 1 . Using the WKB method we manage to get the potential inside the Hamiltonian H , also we evaluate the functional determinant det(H + z 2 ) by means of Zeta regularization, we discuss the similarity of our method to the method applied to get the Zeros of the Selberg Zeta function. In this paper and for simplicity we use units so 2m = 1 =
Eprint Arxiv Math 0607095, Jul 4, 2006
In this paper we present a method to obtain a possible self-adjoint Hamiltonian operator so its e... more In this paper we present a method to obtain a possible self-adjoint Hamiltonian operator so its energies satisfy Z(1/2+iE_n)=0, which is an statement equivalent to Riemann Hypothesis, first we use the explicit formula for the Chebyshev function Psi(x) and apply the change x=exp(u), after that we consider an Statistical partition function involving the Chebyshev function and its derivative so Z=Tr(exp(-BH), from the integral definition of the partition function Z we try to obtain the Hamiltonian operator assuming that H=P^{2}+V(x) by proposing a Non-linear integral equation involving Z(B=-iu) and V(x).
In this paper I give an evaluation of a functional integral by means of a series in functional de... more In this paper I give an evaluation of a functional integral by means of a series in functional derivatives, first of all we propose a differential equation of first order and solve it by iterative methods, to obtain a series for the indefinite integral in one dimension, after that we extend this concept to infinite dimensional spaces, we introduce the functional derivative and propose a functional differential equation for the functional integral which we solve by iteration obtaining a series that includes the n-th order functional derivative dn, this series can be improved by using the Euler transform for alternating series. Also in the second part we study an evaluation of the Path integral using Saddle-point methods, the corrections to the WKB approach that yield to a divergent series are evaluate by means of a Borel transform, to accelerate its convergence we use the Abel-Euler algorithm for power series.
We study a given exponential potential bx ae on the Real half-line which is possible
In this paper we study the methods of Borel and Nachbin resummation applied to the solution of in... more In this paper we study the methods of Borel and Nachbin resummation applied to the solution of integral equation with Kernels K(yx) , the resummation of divergent series and the possible application to Hadamard finite-part integral and distribution theory
In this paper we present a method to deal with divergences in perturbation theory using the metho... more In this paper we present a method to deal with divergences in perturbation theory using the method of the Zeta regularization, first of all we use the Euler-Mc Laurin Sum formula to associate the divergent integral to a divergent sum in the form 1+2^{m}+3^{m}+... After that we find a recurrence formula for the integrals and apply zeta regularization techniques to
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Papers by Jose Javier Garcia Moreta