Physical mathematics in number theory

The paper is about the problem of carefully estimating the bounds, which is sometimes missing in the theoretical physics. Possible consequences of the missing of the bounds is discussed on example of the Riemann zeta function. The text of the paper is based on the drafts of A.A. Karatsuba’s lecture “Physical mathematics in number theory”, devoted to the 85th birthday of academician Vasilii Sergeevich Vladimirov.