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Bump-Ng Theorem

The Bump-Ng theorem states that the zeros of the Mellin transforms of the even-indexed Hermite functions defined below all have real part 1/2. Let H_n be the nth Hermite polynomial, and define the scaled Hermite functions

f_n(x)=2^(-n/2)H_n(sqrt(2pi)x)e^(-pix^2).

Their Mellin transforms on (0,infty) are

M_n(s)=int_0^inftyf_n(x)x^s(dx)/x.

Thus, for even n, every zero s of M_n satisfies R[s]=1/2. Bump and Ng (1986) proved this result. Bump et al. (2000, p. 2) included odd n, following an observation by Vaaler.

See also

Hermite Function, Hermite Polynomial, Mellin Transform

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References

Bump, D.; Choi, K.-K.; Kurlberg, P.; and Vaaler, J. D. "A Local Riemann Hypothesis, I." Math. Z. 233, 1-18, 2000. https://doi.org/10.1007/PL00004786.Bump, D. and Ng, E. K.-S. "On Riemann's Zeta Function." Math. Z. 192, 195-204, 1986. https://doi.org/10.1007/BF01179422.Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, pp. 352 and 391, 2004.

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Bump-Ng Theorem

Cite this as:

Weisstein, Eric W. "Bump-Ng Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Bump-NgTheorem.html

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