I need to find a Dirichlet series f that has the following property.
f is zero in only one point s such that Re(s) > $\sigma_c $.
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1$\begingroup$ $1/\zeta(s){}{}$ $\endgroup$– WojowuCommented May 4, 2019 at 16:04
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7$\begingroup$ Why are people voting to close this? $\endgroup$– LuciaCommented May 4, 2019 at 16:54
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4$\begingroup$ @Lucia I have voted to close because I thought $1/\zeta(s)$ is a (relatively) obvious counterexample, but I have not realized it relies on RH. I have retracted my vote now. $\endgroup$– WojowuCommented May 4, 2019 at 17:51
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3$\begingroup$ Just curious: why did you need to find such an example? $\endgroup$– KConradCommented May 5, 2019 at 10:57
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1$\begingroup$ @Jan-ChristophSchlage-Puchta $\sigma_c$ is a standard notation for the abscissa of convergence, while abscissa of absolute convergence is usually denoted with $\sigma_a$. $\endgroup$– WojowuCommented May 6, 2019 at 11:26
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1 Answer
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That such a Dirichlet series exists was a conjecture of Balazard, which was recently resolved by Hilberdink and Saias. If the Riemann Hypothesis is true, then $1/\zeta(s)$ would provide such an example (with the abscissa of conditional convergence being $1/2$), and the goal was to find an unconditional example.