Related papers: New results based on Riemann hypothesis is tenable
We prove an inequality featuring three well-known functions from analysis, namely the cotangent, the Euler-Riemann zeta function, and the digamma function. Aside from a simple proof of our result, we give a conjectured strengthening. We…
In this paper, a positive answer to the Riemann hypothesis is given by using a new result that predict the exact location of zeros of the alternating zeta function on the critical strip.
As well known, the study of Riemanns zeta function {\zeta}(s) involves the related entire function {\xi}(s). A close relative of {\zeta}(s) is the alternating zeta function {\eta}(s). Similar to {\zeta}(s), also {\eta}(s) has a…
A well-known result, due to Dirichlet and later generalized by de la Vallee-Poussin, expresses a relationship between the sum of fractional parts and the Euler-Mascheroni constant. In this paper, we prove an asymptotic relationship between…
We introduce a new set of prime numbers functions including an exact Generating Function and a Discriminating Function of Prime Numbers neither based on prime number tables nor on algorithms. Instead these functions are defined in terms of…
The Riemann hypothesis (RH) is a long-standing open problem in mathematics. It conjectures that non-trivial zeros of the zeta function all have real part equal to 1/2. The extent of the consequences of RH is far-reaching and touches a wide…
The current research regarding the Riemann zeros suggests the existence of a non-trivial algebraic/analytic structure on the set of Riemann zeros. The duality between primes and Riemann zeta function zeros suggests some new goals and…
Results of a multipart work are outlined. Use is made therein of the conjunction of the Riemann hypothesis, RH, and hypotheses advanced by the author. Let z(n) be the nth nonreal zero of the Riemann zeta-function with positive imaginary…
In this paper we provide a proof of the Riemann Hypothesis by relating the non-trivial zeros of the zeta function to a certain Sturm-Liouville eigenvalue problem on a finite interval.
Some identities for the Riemann zeta-function are proved, using properties of the Mellin transform and M\"untz's identity.
These expository lectures focus on the distribution of zeros of the Riemann zeta function. The topics include the prime number theorem, the Riemann hypothesis, mean value theorems, and random matrix models.
Euler discovered a formula for expressing the value of the Riemann zeta function for all even positive integer arguments. A closed-form expression for the Riemann zeta function for all odd integer arguments, based on the values of the…
A Master equation has been previously obtained which allows the analytic integration of a fairly large family of functions provided that they possess simple properties. Here, the properties of this Master equation are explored, by extending…
This paper studies the connections between the zeros and their distribution functions for two particular Dirichlet $L$ functions: the Riemann zeta function, and the Catalan beta function, also known as the Dirichlet beta function. It is…
One familiar with the Euler zeta function, which established the remarkable relationship between the prime and composite numbers, might naturally ponder the results of the application of this special function in cases where there is no…
In this paper a new integral for the remainder of $\pi(x)$ is obtained. It is proved that there is an infinite set of the formulae containing miscellaneous parts of this integral.
The conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions $f_{house(\alpha)}(z)$ associated with the dynamical zeta functions $\zeta_{house(\alpha)}(z)$ of the…
In the study of Dirichlet series with arithmetic significance there has appeared (through the study of known examples) certain expectations, namely (i) if a functional equation and Euler product exists, then it is likely that a type of…
In 1874, Mertens proved the approximate formula for partial Euler product for Riemann zeta function at $s=1$, which is called Mertens' theorem. In this paper, we generalize Mertens' theorem for Selberg class and show the prime number…
The Mellin transform and several Dirichlet series related with the Riemann zeta function are used to deduce some identities similar to the classical M\"untz formula [4]. These formulas are derived in the critical strip and in the half-plane…