Editing Bernoulli polynomials (section)

==Maximum and minimum==

At higher {{mvar|n}} the amount of variation in <math>B_n(x)</math> between <math>x = 0</math> and <math>x = 1</math> gets large. For instance, <math>B_{16}(0) = B_{16}(1) = {}</math><math> -\tfrac{3617}{510} \approx -7.09,</math> but <math>B_{16}\bigl(\tfrac12\bigr) = {}</math><math>\tfrac{118518239}{3342336} \approx 7.09.</math> {{nobr|[[D.H. Lehmer|Lehmer]] (1940)<ref>{{cite journal |first=D.H. |last=Lehmer |author-link=D.H. Lehmer |year=1940 |title=On the maxima and minima of Bernoulli polynomials |journal=[[American Mathematical Monthly]] |volume=47 |issue=8 |pages=533–538 |doi=10.1080/00029890.1940.11991015 }}</ref>}} showed that the maximum value ({{mvar|M{{sub|n}}}}) of <math>B_n(x)</math> between  {{math|0}}  and  {{math|1}}  obeys
<math display="block">M_n < \frac{2n!}{(2\pi)^n}</math>
unless {{mvar|n}} is {{nobr|{{math|2 modulo 4}},}} in which case
<math display="block">M_n = \frac{2\zeta (n)\,n!}{(2\pi)^n}</math>
(where <math>\zeta(x)</math> is the [[Riemann zeta function]]), while the minimum ({{mvar|m{{sub|n}}}}) obeys
<math display="block">m_n > \frac{ -2 n!}{(2\pi)^n}</math>
unless {{nobr| {{math|1=''n'' = 0 modulo 4 }} ,}} in which case
<math display="block">m_n = \frac{-2 \zeta(n)\,n! }{(2\pi)^n}.</math>

These limits are quite close to the actual maximum and minimum, and Lehmer gives more accurate limits as well.