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In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf (see ) about the rate of growth of the Riemann zeta function on the critical line that is implied by the Riemann hypothesis. It says that, for any ''ε'' > 0, : as ''t'' tends to infinity (see O notation). Since ''ε'' can be replaced by a smaller value, we can also write the conjecture as, for any positive ''ε'', : ==The μ function== If σ is real, then μ(σ) is defined to be the infimum of all real numbers ''a'' such that ''ζ''(''σ'' + ''iT'') = O(''T'' ''a''). It is trivial to check that ''μ''(''σ'') = 0 for ''σ'' > 1, and the functional equation of the zeta function implies that μ(''σ'') = ''μ''(1 − ''σ'') − ''σ'' + 1/2. The Phragmen–Lindelöf theorem implies that μ is a convex function. The Lindelöf hypothesis states μ(1/2) = 0, which together with the above properties of ''μ'' implies that ''μ''(''σ'') is 0 for ''σ'' ≥ 1/2 and 1/2 − σ for ''σ'' ≤ 1/2. Lindelöf's convexity result together with ''μ''(1) = 0 and ''μ''(0) = 1/2 implies that 0 ≤ ''μ''(1/2) ≤ 1/4. The upper bound of 1/4 was lowered by Hardy and Littlewood to 1/6 by applying Weyl's method of estimating exponential sums to the approximate functional equation. It has since been lowered to slightly less than 1/6 by several authors using long and technical proofs, as in the following table: 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lindelöf hypothesis」の詳細全文を読む スポンサード リンク
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