proof of Bohr-Mollerup theorem
Since every term in this series is positive, is logarithmically convex. Furthermore, note that since each term is monotonically decreasing, is a decreasing function of . If for some integer , then we can bound the series term-by-term to obtain
Therefore, as , .
Suppose that is not constant. Then there must exist points and on the real axis such that . Suppose that for definiteness. Since is periodic with period 1, we may assume without loss of generality that . Let denote the second divided difference of :
By our assumptions, . By linearity,
By periodicity, we have
for every integer . However,
As , the right hand side approaches zero. Hence, by choosing sufficiently large, we can make the left-hand side smaller than . For such an ,
However, this contradicts hypothesis 1. Therefore, must be constant. Since , for all , which implies that . In other words, as desired.
|Title||proof of Bohr-Mollerup theorem|
|Date of creation||2013-03-22 14:53:39|
|Last modified on||2013-03-22 14:53:39|
|Owner||Andrea Ambrosio (7332)|
|Last modified by||Andrea Ambrosio (7332)|
|Author||Andrea Ambrosio (7332)|