Sum of reciprocal of integers

[hudson]

given any two numbers a,b and an upper and lower bound for the sum of reciprocals of a certain class of integers between a and b, without any direct calculation how can optimal upper and lower bounds for the number of terms in the sum be found

 

[Eynstone]

I don't see how one could do this without 'any direct calculation'. However, one can find the integral of the characteristic function * (1/x) over [a,b] to estimate the sum.

 

[chaoseverlasting]

What does optimal mean?

 

[dimitri151]

I have to assume that when you say "a certain class of integers" you mean a congruence class, something like the integers in an arithmetic progression a+n b. In that case this is not difficult. Given two numbers A,B, B>A, the number of integers in an arithmetic progression a+n b that are equal to or between two Numbers A, B , and therefore the number of terms in the required sum is [(B-a)/b]-[(A-a)/b] where [] denotes the integer part of the quantity in the brackets. A more interesting question is: given the integers A, B and an arithmetic progression (a,b) to come up with upper and lower bounds for the sum of the reciprocals of the terms in the arithmetic progression that are between or equal to A,B.

 

[blue_raver22]

?

To find optimal upper and lower bounds for the number of terms in the sum of reciprocals of integers between a and b, one could use mathematical techniques such as the Cauchy-Schwarz inequality or the Euler-Maclaurin formula. These methods involve using the properties of the sum of reciprocals and the values of a and b to derive an expression for the upper and lower bounds. Another approach could be to use computer algorithms or simulations to test different combinations of integers within the given bounds and determine the maximum and minimum number of terms needed to reach the desired sum. Additionally, one could also analyze the pattern of the sum of reciprocals for different classes of integers and use that information to estimate the optimal bounds for the number of terms. Ultimately, finding the optimal upper and lower bounds would require a combination of mathematical reasoning, computational methods, and pattern analysis.