PRISE project 2008
KAM: Dirichlet series
Strange Sum
Office: SciCtr 434
Example geometric series: What is
    1 + 2 + 4 + 8 + 16 + ...   ?
Mathematicians have learned to make sense of this using the concept of analytic continuation. A geometric series S(a) = 1+a+a2 + .... has for |a| smaller than 1 a closed solution formula S = 1/(1-a). This formula makes sense for any complex number a different from 1. It also works for a=2 and gives as the answer the above problem the value -1. We say S=-1 is the regularized value for the above sum 1+2+4+8+16 +... This value has been obtained by analytic continuation.

Example Fourier series: it is considered a historical curiosity but already Fourier tackled the question to make sense of nonconvergent series. We know
   x = 2 sin(x) - 2 sin(2x)/2 + 2 sin(3x)/3 - ... 
If we differentiate this series formally, we obtain
   1 = 2 cos(x) - 2 cos(2x) + 2 cos(3x) - 2 cos(4x) ...
and especially
   1-1+1-1+1-1 ... = 1/2
This is actually the value of the Dirichlet eta function (also called alternating Riemann zeta function)
  E(s) = 1 - 2-s + 3-s - ....
at s=0. Example Dirichlet series: Not only geometric series and Fourier series also the Riemann zeta function
  Z(s) = 1 + 2-s + 3-s + .... 
allows an analytic continuation beyond the place, where the sum converges. For example
  Z(-3) = 1 + 8 + 27 + 64 + 125 + ...
can be assigned the value Z(-3) = 1/120. More generally, Hurwitz zeta functions
  Z(s,b) = (1+b) + (2+b)-s + (3+b)-s  + ... 
have an analytic continuation except for s=1.

Example Almost periodic Dirichlet series: In our research, John Lesieutre and I looked at generalized Riemann zeta functions
   Z(s,g,a) = g(a) + g(2a) 2-s +  g(3a) 3-s + ...  ,
where g(x) is a 1 periodic function and a is a real number. If a=p/q is rational, then one can write the series as a sum of Hurwitz zeta functions for which there is an analytic continuation except at a pole at 1. If g is odd, then the pole disappears in the sum.

We proved that if g is realanalytic (g(x) has a convergent Taylor expansion at each point x) and g(x) averaged over 0 to 1 is zero and a is Diophantine (there exists r larger than 1 and a positive constant C such that for all rational numbers p/q, we have |a-p/q| > C/q1+r, then Z(s,g,a) has an analytic continuation to the entire complex plane. This implies for example that there is a regularized value for
   sin(sin(pi sqrt(2)) + sin(sin(2 pi sqrt(2)) + sin(sin(3 pi sqrt(2)) +  ... 
Many open questions remain. We do not know whether it is possible to assign a regularized value to
   sin(sin(e sqrt(2)) + sin(sin(2 e sqrt(2)) + sin(sin(3 e sqrt(2)) +  ...
for example because we do not know whether e sqrt(2)/pi is Diophantine and what happens in the non Diophantine case. Nor do we know whether it is possible to give a regularized value to
   |sin(sin(pi sqrt(2))| + |sin(sin(2 pi sqrt(2))| + |sin(sin(3 pi sqrt(2))| +  ...
because g(x) = |sin(sin(pi x))| is not real analytic.

To the history of divergent series: while Newton and Leibniz have used infinite series systematically, there was little use of divergent series before Euler, who used divergent series regularly and extensively. Mathematics moved then away from divergent series towards orthodoxy imposed by Cauchy, Abel, d'Alembert, Laplace or Lagrange and Cauchy (See Hardy: "Divergent Series"). With Riemann and the development of stronger analytic methods, the work with divergent series picked up again and became an important part of modern mathematics. They appear also in concrete problems of mathematical physics. For example, its useful to have a determinant for the Laplacian L of a manifold. If the eigenvalues of L are li, then define det(L) = exp(f'(0)), where f is the analytic continuation of
f(s) = 1/l1s + 1/l2s + ...


When we look at a series, the series itself only describes part of the actual function. Analytic continuation and summation techniques reveal much more of the actual function. A local power series only sees a small part of a much larger world, the Riemann surface defined by the function.
Questions and comments to knill@math.harvard.edu
Oliver Knill | Department of Mathematics | Harvard University