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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:
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