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What does this mean: ##f(t) \ll_\epsilon (g(t))^\epsilon## ?
Ref: Beginning of the introduction here.
Ref: Beginning of the introduction here.
Double less-than sign
The double less-than sign, <<, may be used for an approximation of the much-less-than sign (≪) or of the opening guillemet («). ASCII does not encode either of these signs, though they are both included in Unicode.
Other consequences
Denoting by pn the n-th prime number, a result by Albert Ingham shows that the Lindelöf hypothesis implies that, for any ε > 0,
if n is sufficiently large. However, this result is much worse than that of the large prime gap conjecture.
jedishrfu said:I couldn't access the researchgate link as it may be a paywall.
Let s=σ+it be a complex variable and ζ(s)the Riemann zeta-function. The classical Lindelof hypothesis asserts that ##\zeta(\sigma+it) \ll_\epsilon (1+|t|)^\epsilon## for every positive ##\epsilon##.
I've never seem that notation before and I don't know what it, ##\ll_\epsilon##, means. The Wikipedia page, https://en.wikipedia.org/wiki/Lindelöf_hypothesis, says something different.Let s=σ+it be a complex variable and ζ(s)the Riemann zeta-function. The classical Lindelof hypothesis asserts that ##\zeta(\sigma+it) \ll_\epsilon (1+|t|)^\epsilon## for every positive ##\epsilon##.
Yes, in most cases. The inequality ##\epsilon>0## is almost universally meant to convey the idea the ##\epsilon## is a small, positive number.Swamp Thing said:Another quick question: When we read "for ##\epsilon>0## ", does it imply ##\epsilon<1## even if that's not stated?
This doesn't make any sense. The "big O" business is applied to some function; e.g., O(f(x)).Swamp Thing said:I mean specifically when talking about big-O to the power of ##\epsilon## ?.
Sorry, that was sloppy language. I meant big-O applied to some f(x) that involves something raised to ##\epsilon##Mark44 said:This doesn't make any sense. The "big O" business is applied to some function; e.g., O(f(x)).
Subscripted notation in mathematics signifies a function within another function. In this case, f(t) is the main function and g(t) is a sub-function. The epsilon symbol (ε) indicates that the output of g(t) is an element of the set of possible values for f(t).
Regular function notation, such as f(x), indicates that the input is a variable (x) and the output is a single value (f(x)). Subscripted notation adds an additional layer of complexity by indicating that the output of the sub-function (g(t)) is a possible value for the main function (f(t)).
The epsilon symbol (ε) represents the relationship between the main function and the sub-function. It indicates that the output of the sub-function is a possible value for the main function.
An example of this notation could be f(x) = x2 subscripted to g(x) = 2x. In this case, the output of g(x) (2x) is an element of the set of possible values for f(x) (x2). So, f(x) = 4 subscripted to g(x) = 2 is a valid example.
This notation is commonly used in mathematical models and equations to represent complex relationships between different variables. It allows scientists to express the idea that the output of one function is a possible value for another function, which is often the case in scientific experiments and studies.