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On the Tauberian constant in the Ikehara theorem. (English) Zbl 1074.40500
Summary: The Ikehara theorem says: Let the Laplace transform $$f(s)$$ of a nonnegative nondecreasing function $$A(x)$$ defined for $$x\in \langle 0; + \infty )$$ converge in the halfplane $$\text{Re} \;s>0$$ and $$g(s) = f(s)-\frac 1{s-1}$$ be analytic in the halfplane $$\text{Re} \;s\geq 0$$. Then $$e^{-x}A(x) - 1 = o(x)$$, $$x\rightarrow +\infty$$. The only theorem of the paper generalizes the main term 1 to some polynomial and the remainder term $$o(x)$$ to $$O(x^{-n})$$ assuming the function $$g^{(n)}(it)$$ integrable, $$n\in \mathbb N$$, where $$g(s)$$ is the regular part of $$f(s)$$ in the point $$s_0 = 1$$. As the constant in $$O$$ is a multiple of the norm of $$g^{(n)}(it)$$, the author uses the uncertain adjective Tauberian. The theorem contains some inconvenient assumptions. The theorem is used to obtain the remainder term in the P.N.T. in the form $$O(x \exp p (-c\log ^{\frac 1{11}}x))$$, $$x\rightarrow +\infty$$. By analytical methods an essentially better estimation of this remainder term can be obtained.

##### MSC:
 40E05 Tauberian theorems 11N05 Distribution of primes 11M45 Tauberian theorems
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##### References:
 [1] Chandrasekharan K.: Introduction to Analytic Number Theory. Springer Verlag, 1968. · Zbl 0169.37502 [2] Čížek J.: On the proof of the prime number theorem. Čas. pěst. mat. 106 (1981), 395-401. [3] Rudin W.: Real and Complex Analysis. New York, 1974. · Zbl 0278.26001 [4] Subchankulov M. A.: Tauberovy teoremy s ostatkom. Nauka, Moskva, 1976. [5] Walfisz A.: Weyl’sche Exponentialsummen in der neuere Zahlentheorie. Berlin, 1963. · Zbl 0146.06003
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