also presented in [@BR05; @Dr00; @Kh11(2); @Kh11(3)] . The above estimates were sharpened in the papers by A. Klein [@Klein; @Klein1]. In 2001 Ye. S. Chibrikov proved in [@Ch01] that $\Ht(4,l) \geqslant  (7k^2-2k).$ M. I. Kharitonov in [@Kh11; @Kh11(2); @Kh11(3)] obtained estimates for the structure of piecewise periodicity. In 2011 A. A. Lopatin [@Lop11] obtained the following result:

Let $C_{n,l}$ be the nilpotency degree of a free $l$-generated algebra satisfying $x^n=0$, and let $p$ be the characteristic of the ground field of the algebra, greater than $n/2.$ Then $$(1): C_{n,l}<4\cdot 2^{n/2} l.$$

By definition $C_{n,l}\leqslant \Psi(n, n, l).$ Observe that for small $n$ the estimate (1) is smaller than the estimate $\Psi(n, n, l)$ established in this paper but for growing $n$ the estimate $\Psi(n,n,l)$ is asymptotically better than (1).

Ye. I. Zelmanov has put the following question in the Dniester Notebook [@Dnestrovsk] in 1993:

Let $F_{2,m}$ be the free $2$-generated associative ring with identity $x^m=0.$ Is it true that the nilpotency class of $F_{2,m}$ grows exponentially in $m?$

Our paper answers Ye. I. Zelmanov’s question as follows: the nilpotency class in question grows subexponentially.

The results obtained
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[**The main result**]{} of the paper is as follows:

\[c:main2\] The 