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D**u
发帖数: 204 | 1 鉴于最近是数论热，那我讲一个非常有趣的数论中的反例。
来自Notices Nov. 2011 "Exploratory Experimentation and computation".
The formula goes with the name "sophomore's dream", where
we want to see if Sum_p = Integral_p is always true for all prime P's.
For a prime number P, let
Sum_p := sum_{n=−∞}^{n=∞} sinc(n) sinc(n/3) sinc(n/5) sinc(n/7)· ·
· sinc(n/p)
and
Integral_p := int_{-∞}^{∞} sinc(x) sinc(x/3) sinc(x/5) sinc(x/7) · · ·
sinc(x/p) dx,
where the denominators range over the odd primes,
Provably, the following is true: The “sum equals integral” identity for
"Sum_p = Integral_p" remains valid at least for p among the
first 10176 primes but stops holding after some larger prime,
and thereafter the “sum less the integral” is strictly positive,
but they always differ by much less than one part in a googolplex = 10^100.
An even stronger estimate is possible assuming the
generalized Riemann hypothesis. | z***e
发帖数: 5600 | 2 神得不可思议
·
【在 D**u 的大作中提到】
: 鉴于最近是数论热，那我讲一个非常有趣的数论中的反例。
: 来自Notices Nov. 2011 "Exploratory Experimentation and computation".
: The formula goes with the name "sophomore's dream", where
: we want to see if Sum_p = Integral_p is always true for all prime P's.
: For a prime number P, let
: Sum_p := sum_{n=−∞}^{n=∞} sinc(n) sinc(n/3) sinc(n/5) sinc(n/7)· ·
: · sinc(n/p)
: and
: Integral_p := int_{-∞}^{∞} sinc(x) sinc(x/3) sinc(x/5) sinc(x/7) · · ·
: sinc(x/p) dx,
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