From Pythagorean theorem to Riemann's conjecture-Riemann zeta function as a biological system

Riemann ’ s zeta function and the prime number theorem

16 Riemann’s zeta function and the prime number theorem We now divert our attention from algebraic number theory to talk about zeta functions and L-functions. As we shall see, every global field has a zeta function that is intimately related to the distribution of its primes. We begin with the zeta function of the rational field Q, which we will use to prove the prime number theorem. We will ne...

An Effective Universality Theorem for the Riemann Zeta Function

Let 0 < r < 1/4, and f be a non-vanishing continuous function in |z| ≤ r, that is analytic in the interior. Voronin’s universality theorem asserts that translates of the Riemann zeta function ζ(3/4 + z + it) can approximate f uniformly in |z| < r to any given precision ε, and moreover that the set of such t ∈ [0, T ] has measure at least c(ε)T for some c(ε) > 0, once T is large enough. This was...

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...

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 ...

Mollifying the Riemann Zeta-function