Infinite arithmetic sequence

In mathematics, an infinite arithmetic sequence is an infinite sequence whose terms are in an arithmetic progression. Every term after the first of such a sequence is the arithmetic mean of the two neighboring terms. These sequences are studied in number theory, where Dirichlet's theorem shows that given any two coprime positive integers a and b, the infinite arithmetic sequence whose terms are $an+b$ contains infinitely many prime numbers. When the terms of the sequence are summed, the result is an infinite arithmetic series, for example 1 + 1 + 1 + 1 + · · · and 1 + 2 + 3 + 4 + · · ·. The general form for an infinite arithmetic series is

$\sum _{n=0}^{\infty }(an+b).$

If a = b = 0, then the sum of the series is 0. If either a or b is nonzero, then the series diverges and has no sum in the usual sense.

Zeta regularization

The zeta-regularized sum of an arithmetic series of the right form is a value of the associated Hurwitz zeta function,

$\sum _{n=0}^{\infty }(n+\beta )=\zeta _{\mathrm {H} }(-1;\beta ).$

Although zeta regularization sums 1 + 1 + 1 + 1 + · · · to ζ_R(0) = −1/2 and 1 + 2 + 3 + 4 + · · · to ζ_R(−1) = −1/12, where ζ is the Riemann zeta function, the above form is not in general equal to

$-{\frac {1}{12}}-{\frac {\beta }{2}}.$

References

• Brevik, I.; Nielsen, H. B. (February 1990). "Casimir energy for a piecewise uniform string". Physical Review D. 41 (4): 1185–1192. doi:10.1103/PhysRevD.41.1185.
• Elizalde, E. (May 1994). "Zeta-function regularization is uniquely defined and well". Journal of Physics A: Mathematical and General. 27 (9): L299–L304. doi:10.1088/0305-4470/27/9/010. (arXiv preprint)
• Li, Xinzhou; Shi, Xin; Zhang, Jianzu (July 1991). "Generalized Riemann ζ-function regularization and Casimir energy for a piecewise uniform string". Physical Review D. 44 (2): 560–562. doi:10.1103/PhysRevD.44.560.

See also

• 1 − 2 + 3 − 4 + · · ·

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