Let L(s)=∑ a_n n^-s be an ordinary Dirichlet series with entire continuation of order at most 1, and let mu(sigma) denote its order function on vertical lines. Refined Sharp Kahane conjecture. For every sigma satisfying mu(sigma)≥ 1, mu bigl(sigma+mu(sigma)bigr)=0. Equivalently, on any interval where mu≥ 1, the slope of mu is at most -1. The unrestricted relation mu(sigma+mu(sigma))=0 would imply the Lindelöf hypothesis for the Riemann zeta function, so the paper isolates this restricted form as a potentially accessible statement; its status is presented as open.

The refined Sharp Kahane conjecture for ordinary Dirichlet series

The refined conjecture asks whether an ordinary Dirichlet series with entire continuation of order at most $1$ satisfies $\mu(\sigma+\mu(\sigma))=0$ whenever $\mu(\sigma)\ge 1$ . Harald Bohr posed the original problem in 1952; Kahane’s later counterexamples concern the unrestricted version and are not entire.

Known results

Bohr, 1952: the unrestricted relation for the Dirichlet eta function would imply the Lindelöf hypothesis for the Riemann zeta function.
Kahane, 1989: constructed unrestricted counterexamples admitting continuation only to a half-plane.
Theorem 5.5: for entire Dirichlet series of horizontal and vertical degree $1$ , intervals where $\mu\ge 1$ have slope in $[-1,-2/3]$ .

March 2026 partial restrictions

The paper Bohr’s Last Problem Under the Entirety Hypothesis labels the refined statement Conjecture 5.3 and explicitly leaves it open. Its remaining gap is to exclude slopes in $(-1,-2/3]$ and thereby force slope $-1$ ; no proof, counterexample, or claimed settlement was found.

Current status (as of August 2026): The refined conjecture remains open; the degree- $1$ slope restriction is known, but no proof or counterexample to the stated conjecture has been reported.

The refined Sharp Kahane conjecture for ordinary Dirichlet series

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Progress summary

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