Recent content by mathslover

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    Summation involving von Mangoldt function

    I have tried numerical calculation and the sum seems to converge to ~ -1.16 Can we approach the problem from Zeta function? -Ng
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    Summation involving von Mangoldt function

    Hi CRGreatHouse, In your post 1673, the summation on LHS runs from n=2 to infinity, (n=2,3,4,5,6,7,8,...) But the summation on RHS runs over all primes.(p=2,3,5,7,...) From the definition of von Mangoldt function,when n=6,10,12,14,15,18,... , the summand became (-1/n) whenever n...
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    Summation involving von Mangoldt function

    when n is a prime or prime power, the summation is okay. but suppose when n=6, then the sum is (von Mangoldt(6) -1)/6 , which is = -1/6, as n runs from 2 to infinity,can we settle the problem of convergency or divergency? -Ng
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    Summation involving von Mangoldt function

    hi mhill, can you prove that the series is divergent? -Ng
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    Two Integrals

    For the first integral , it can be shown to be = Gamma''(1) and - Euler's Constant = Gamma'(1) where Gamma(x)=Gamma Integral I just don't know how to use the above facts.
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    Two Integrals

    I have tried many hours on the following integrals and would appreciate any help from you. 1. Int{x=0 to infinity}(exp(-x)*Ln(x)*Ln(x)dx) 2. Int{x=0 to Infinity}(exp(-x*x)*Ln(x)dx) Any idea guys?
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    Summation involving von Mangoldt function

    Please help me in solving the problem, find the sum Sum{r=2 to infinity} (von Mangoldt(r)-1)/r Your help is appreciated.
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    Strange Incredible Identity

    How can we proceed to prove the following identity ?
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    A Difficult Integral

    Leafing through "Treatise on Integral Calculus Vol. 2 --Joseph Edwards (1922)",I found Wolstenholme had solved the above problem nicely as follow: -Ng
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    A Difficult Integral

    How should we proceed to find the definite integral Int[ log(sin(x))*log(cos(x)) ,{x,0,pi/2} ] ? mathslover
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    Integral Calculation

    The beautiful point about this calculation is that it is applicable for all positive integers. Define I(n)=Int((log(sin(x)))^n, {x=0 to pi/2}) then it can be shown that I(0) = pi/2 I(1) = -(pi/2)*log(2) I(2) = (pi^3)/(24) + (pi/2)*(log(2))^2 and I(3) is a function of...
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    Area of polar curve

    when the angle changed from 0 to pi/2 ,you DON'T get the whole curve! Try drawing out the curve.Note, cos(5*angle) must be > 0 to give a real point on the curve. when angle=0, r=3 when angle=18 degree, r=0 hope this is helpful
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    Integral Calculation

    Reading through "Ramanujan's notebook Part 2" and "Collected Papers of Ramanujan ", I chanced upon an entry which solved my problem beautifully. I just wish that much more can be learned from Ramanujan's work. -mathslover
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    Integral Calculation

    I would like to find the definite integral of (log(sin(x)))^2 under the interval (pi/2,0) Numerical integration can only give a numerical answer I would like to find the above integral in terms of well-known constants -Mathslover
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    Integral Calculation

    I am sorry , the integrand should be (log(sin(x)))^2 and the interval is (Pi/2, 0) . -Mathslover
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