Ikehara-type theorem involving boundedness
Abstract
Consider any Dirichlet series sum an/nz with nonnegative coefficients an and finite sum function f(z)=f(x+iy) when x is greater than 1. Denoting the partial sum a1+...+aN by sN, the paper gives the following necessary and sufficient condition in order that (sN)/N remain bounded as N goes to infinity. For x tending to 1 from above, the quotient q(x+iy)=f(x+iy)/(x+iy) must converge to a pseudomeasure q(1+iy), the distributional Fourier transform of a bounded function. The paper also gives an optimal estimate for (sN)/N under the "real condition" that (1-x)f(x) remain bounded as x tends to 1 from above.
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