We expose techniques that permit to approximate the Riemann Zeta function in different context. The motivation does not restrict only to the numerical verification of the Riemann hypothesis (see Numerical computations about the zeros of the zeta function) : in [5] for example, Lagarias and Odlyzko obtained the best asymptotic known method to compute π(x), the number of primes less than x, thank...

The “hybrid” moments Z 2T T |ζ( 1 2 + it)| „ Z t+G t−G |ζ( 1 2 + ix)| dx m dt of the Riemann zeta-function ζ(s) on the critical line Re s = 1 2 are studied. The expected upper bound for the above expression is Oε(T G). This is shown to be true for certain specific values of k, l,m ∈ N, and the explicitly determined range of G = G(T ; k, l,m). The application to a mean square bound for the Melli...

We prove the Chern-Weil formula for SU(n + 1)-singular connections over the complement of an embedded oriented surface in smooth four manifolds. The expression of the representation of a number as a sum of nonvanishing squares is given in terms of the representations of a number as a sum of squares. Using the number theory result, we study the irreducible SU(n+ 1)-representations of the fundame...

Let P be the set of all primes and ψ(n) = n ∏ n∈P,p|n(1 + 1/p) be the Dedekind psi function. We prove that, under the assumption that the n-th prime pn is always larger than the first Chebyshev function θ(pn), the Riemann hypothesis is satisfied if and only if f(n) = ψ(n)/n − e log logn < 0 for all integers n > n0 = 30 (D), where γ ≈ 0.577 is Euler’s constant. This inequality is equivalent to R...

According to my first teacher Gustave Choquet one does, by openly facing a well known unsolved problem, run the risk of being remembered more by one’s failure than anything else. After reaching a certain age, I realized that waiting “safely” until one reaches the end-point of one’s life is an equally selfdefeating alternative. In this paper I shall first look back at my early work on the classi...

In [1] the author proposed two new results concerning prime zeta function and Riemann but they turn out to be wrong. present paper we provide their correct form.

Techniques for using the Monte Carlo method to evaluate fuzzy Riemann integrals and improper fuzzy Riemann integrals are proposed in this paper. Owing to the α-level set of the (improper) fuzzy Riemann integral being the closed interval whose end points are the classical (improper) Riemann integrals, it is possible to invoke the Monte Carlo method to approximate the end points of the α-level cl...

This paper is devoted to the study of establishing sufficient conditions for existence and uniqueness of positive solution to a class of non-linear problems of fractional differential equations. The boundary conditions involved Riemann-Liouville fractional order derivative and integral. Further, the non-linear function $f$ contain fractional order derivative which produce extra complexity. Than...