An orthogonal projection related to the Riemann zeta-function

A Closure Problem Related to the Riemann Zeta-function.

It is rather obvious that any property of the Riemann zeta-function may be expressed in terms of some other property of the function p(x) defined as the fractional part of the real number x, i.e., x = p(x) mod 1. This note will deal with a duality of the indicated kind which may be of some interest due to its simplicity in statement and proof. In the sequel, C will denote the linear manifold of...

An amortized-complexity method to compute the Riemann zeta function

A practical method to compute the Riemann zeta function is presented. The method can compute ζ(1/2 + it) at any T 1/4 points in [T, T + T 1/4] using an average time of T 1/4+o(1) per point. This is the same complexity as the Odlyzko-Schönhage algorithm over that interval. Although the method far from competes with the Odlyzko-Schönhage algorithm over intervals much longer than T 1/4, it still h...

q-Riemann zeta function

We consider the modified q-analogue of Riemann zeta function which is defined by ζq(s)= ∑∞ n=1(qn(s−1)/[n]s), 0< q < 1, s ∈ C. In this paper, we give q-Bernoulli numbers which can be viewed as interpolation of the above q-analogue of Riemann zeta function at negative integers in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Also, we will treat some...

Mollifying the Riemann Zeta-function

Lagrangians with Riemann Zeta Function

We consider construction of some Lagrangians which contain the Riemann zeta function. The starting point in their construction is p-adic string theory. These Lagrangians describe some nonlocal and nonpolynomial scalar field models, where nonlocality is controlled by the operator valued Riemann zeta function. The main motivation for this research is intention to find an effective Lagrangian for ...