Regularization for zeta functions with physical applications I

A proof for the original Riemann hypothesis is proposed based on the infinite Hadamard product representation for the Riemann zeta function and later generalized to Dirichlet L-functions. The extension of the hypothesis to other functions is also discussed.

A mathematical proof is only true if the proof can be reproducible, and perhaps by alternative means than that employed in the first proof. A proof of the Riemann Hypothesis should be generalizable because there exists zeta functions such as the Dedekind zeta function, Dirichlet series, generalized zeta functions, and L-Functions. Although we do not consider here the generalized zeta functions, it is my goal to show the reader that this proof of the Riemann Hypothesis is gene...

This review article brings forth some recent results in the theory of the Riemann zeta-function $qzeta(s)$.

We present some novelties on the Riemann zeta function. Using the analytic continuation we created for the polylogarithm, $\mathrm{Li}_{k}(e^{m})$, we extend the zeta function from $\Re(k)>1$ to the complex half-plane, $\Re(k)>0$, by means of the Dirichlet eta function. More strikingly, we offer a reformulation of the Riemann hypothesis through a zeta's cousin, $\varphi(k)$, a pole-free function defined on the entire complex plane whose non-trivial zeros coincide with those o...

This paper considers some infinite series involving the Riemann zeta function.

In this article, our primary objective is to provide an extensive introduction to the Riemann Zeta Function $\zeta(s)$, an integral part in the study different Analytic aspects relevant to the proof of the famous "Prime Number Theorem" (PNT). The whole paper comprises of three mutually exclusive parts. In the initial sections, we define all the necessary terminologies and results handpicked from the areas related to Analytic Number Theory and Analysis of Complex Numbers, wher...

In the present manuscript, we study analytic properties of zeta functions defined by partial Euler products.

We present another expression to regularize the Euler product representation of the Riemann zeta function. % in this paper. The expression itself is essentially same as the usual Euler product that is the infinite product, but we define a new one as the limit of the product of some terms derived from the usual Euler product. We also refer to the relation between the Bernoulli number and $P(z)$, which is an infinite summation of a $z$ power of the inverse primes. When we apply...

We present a conjecture about the asymptotic representation of certain series. The conjecture implies the Riemann hypothesis and it would also indicate the simplicity of the non-trivial zeros of the zeta-function.

In this article, with a new approach, which is not discussed in the literature yet, the limit of the Riemann zeta function or Euler-Riemann zeta function is approximately explored by applying Dirichlet's rearrangement theorem for absolutely convergent series to the Riemann zeta function by rearranging its terms as geometric series for sufficiently large $n$. The limit of the Riemann zeta function or Euler-Riemann zeta functions, $\lim_{n\to\infty} \zeta(z)$, is first time exp...