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2h
comment series involving Catalan and Zeta
Thanks for the edit @Lucian! (concerning ^' I think it worked until around one and half year earlier... possibly a Mathjax evolution since I still use the same old safari browser). Cheers,
13h
comment Help to solve complex equation related to the Gamma function
@EddyKhemiri: I added the corresponding handling for the $\theta$ function to clarify things further. Cheers,
13h
revised Help to solve complex equation related to the Gamma function
Handling too the argument of R-S theta
1d
comment Help to solve complex equation related to the Gamma function
@EddyKhemiri: Well, from the link at the end of my answer we see that the asymptotic distribution of zeros is the same as the distribution of zeros of the Riemann-Siegel $\theta$ function but this seems to be all we can tell at this point : the actual positions are regular only for $\Gamma$ and not for $\zeta$! B.t.w. could you verify that the values I provided verify $\;\arg\left(\zeta\left(\frac 12+iy\right)\right)=k\pi/2$. I think that there is a relation between the zeros here and those of R-S $\theta$ but I don't know it yet...
1d
comment Help to solve complex equation related to the Gamma function
Glad you liked it @EddyKhemiri ! It would be very pleasant to see a simpler formula for the roots (I only provided an approximation...) since reverting $\Gamma$ is not that easy... Wishing you fun with all that in the meantime,
1d
answered Help to solve complex equation related to the Gamma function
2d
awarded  calculus
Dec
15
answered How to integrate $\ln \big( b + \sqrt{b^2 + c^2 + x^2}\,\big)$?
Dec
15
awarded  number-theory
Dec
14
comment Proper Bernoulli Function Generating Function
Of course D. Knuth's 1993 paper "Johann Faulhaber and sums of powers" shoudn't be missed! Excellent explorations,
Dec
14
comment Proper Bernoulli Function Generating Function
Concerning approximations this paper may too be of interest.
Dec
14
comment Proper Bernoulli Function Generating Function
@frogeyedpeas: Ok so you want to generalize the Faulhaber's formula to non integer powers. This finite sum may in fact be rewritten as a difference of two Hurwitz zeta functions as shown in this paper or in MathWorld's "Power sum" : $$\sum_{i=0}^{x}i^s=\zeta(-s,1)-\zeta(-s,x+1)$$ (let's be clear : this is a mere rewriting of your sum as you may see from the zeta definition)
Dec
14
comment Proper Bernoulli Function Generating Function
Multiplying this by a Bernoulli number will give you something rather more complicated than the simple $B_i\,x^i$ from my answer I fear... For some equalities and generating functions for central binomial coefficients you may look at this link but here we have the additional Bernoulli coefficient. I didn't verify your last equality yet (I have to go...) but the expansion of sqrt will usually provide these central binomial terms.
Dec
14
comment Proper Bernoulli Function Generating Function
@frogeyedpeas: Well $\;\Gamma(1/2)=\sqrt{\pi}\,$ and $\;\Gamma(1/2-i)=\dfrac{(-4)^i\,i!}{(2i)!}\sqrt{\pi}\;$ from gamma's properties (btw the last formula for $\;\Gamma(1/2-i)$ is wrong ; it should be a Pochhammer symbol I think!) so that $$\frac{(-1)^i\;\Gamma\bigl(\frac{3}{2}\bigr)}{\Gamma(i)\Gamma\bigl(\frac{3}{2}-‌​i\bigr)}=\frac {\frac 12 (2i)!}{(i-1)!\bigl(\frac{1}{2}-i\bigr)\;4^i\,i!}=\frac{i}{(2i-1)\,4^i}\binom{2i}‌​{i}$$
Dec
13
revised How many consecutive squares can be subtracted from a number?
added 9 characters in body
Dec
13
answered How many consecutive squares can be subtracted from a number?
Dec
12
comment Riemann zeta function and modulus
Thanks to take care of this old thread @mike and for the references. I don't have the 1965 Spira paper (I got the others) but found this recent one by Trudgian (I got earlier too the papers from Nazardonyavi and Yakubovich). I won't hold my breath to prove RH this way but wish you fun trying it this way anyway!
Dec
12
revised Proper Bernoulli Function Generating Function
added 55 characters in body
Dec
12
answered Proper Bernoulli Function Generating Function
Dec
11
answered About the Erdős-Borwein Constant