Uniform approximation of some Dirichlet series by partial products of Euler type

In the present work we show that the Dirichlet series with the Euler product having analytical continuation to the critical strip without singularities, in some natural conditions, can be approximated by partial products of Euler type in the critical strip, if the primes over which are taken the products are distributed by a suitable way. The family of such series includes many of widely used Dirichlet series as the zeta-function, Dirichlet L-functions and etc. As a consequence the analog of the Riemann Hypothesis for such series is proven.

💡 Research Summary

The paper tackles the long‑standing problem of approximating Dirichlet series that possess an Euler product and admit analytic continuation into the critical strip (0 < Re s < 1) without singularities. The author’s main claim is that, under a set of “natural” conditions, such series can be uniformly approximated in the critical strip by finite Euler‑type products, provided the primes entering the product are selected according to a specially designed distribution. The work is positioned as a unifying framework that covers the Riemann zeta‑function, Dirichlet L‑functions, and many other commonly used Dirichlet series. As a corollary, the author asserts that an analogue of the Riemann Hypothesis (RH) holds for all series within this class.

The technical core begins by fixing a Dirichlet series
(F(s)=\sum_{n\ge1}a_n n^{-s}=\prod_{p}(1-b_p p^{-s})^{-1})
with coefficients (a_n, b_p) satisfying (|b_p|\le1). The series is assumed to be analytic throughout the critical strip and free of poles, a hypothesis that is automatically satisfied by the Riemann zeta‑function (after removing its simple pole at s = 1) but is restrictive for general L‑functions that possess non‑trivial zeros.

To construct the approximating finite product, the author introduces a “prime selection function” (\phi(p)\in{0,1}). The set of primes used in the partial product is (\mathcal{P}X={p:,p\le X,\ \phi(p)=1}). The function (\phi) is required to satisfy a density condition: the weighted sum (\sum{p\le X}\phi(p)p^{-\sigma}) must converge uniformly for (\sigma) in any compact sub‑interval of (0,1). In practice this is achieved by imposing a smooth density function (\delta(p)) that mimics the prime number theorem’s main term (1/\log p) while allowing controlled deviations.

The main theorem states that for any compact set (K\subset{s:0<\Re s<1}) and any (\varepsilon>0) there exists an (X_0) such that for all (X\ge X_0)
\