discussed in detail.'
author:
- '**Yaroslav D. Sergeyev[^1]**'
bibliography:
- 'XBib\_accuracy.bib'
title: |
    **On accuracy of mathematical languages used\
    to deal with the Riemann zeta function and\
    the Dirichlet eta function**
---

Infinite and infinitesimal numbers and numerals; accuracy of mathematical languages; Riemann zeta function; Dirichlet eta function; divergent series.

Introduction {#s1_riem}
============

The Riemann zeta function (s)=\_[u=1]{}\^ \[Riemann\_zeta\] defined for complex numbers $s$ is one of the most important mathematical objects discovered so far (see [@Conrey; @Edwards]). It has been first introduced and studied by Euler (see [@Euler_1; @Euler_2; @Euler_3; @Euler_4]) who proved the famous identity \_[u=1]{}\^ = \_[p ]{} \[Riemann\_primes\] being one of the strongest sources of the interest to the Riemann zeta function. Many interesting results have been established for this function (see [@Conrey; @Edwards] for a complete discussion). In the context of this paper, the following results will be of our primary interest.

It is known that $\zeta(s)$ diverges on the open half-plane of $s$ such that the real part $\Re(s) < 1$. Then, for $s=0$ the following value is attributed to $\zeta(0)$: (0)=\_[u=1]{}\^ = 1+1+1+…= -. \[1+1\] It is also known that the Riemann zeta function has the following relation (s) = (1- 2\^[1-s]{})(s) \[Riemann\_relation\] 