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The classical Euler beta function can be defined by $$ B(p,q)=\int_0^1t^{p-1}(1-t)^{q-1}\operatorname{d\!}t $$ for $\Re(p),\Re(q)>0$. The beta function and the classical Euler gamma function $\...

I guess for the modified Bessel funcion $K_0(z)$, $$\sum_{n=1}^\infty K_0(s\, n) \sim \frac{-2\log 2 - \log \pi + \gamma}{2} + \frac{\log s}{2} + \frac{\pi}{2\, s}, \quad s\to 0^+,$$ if taking $$\...

Does $$\frac{1}{N^2}\sum _{d=1}^N \log d \sum _{n=1}^{N/d} \frac{\phi(n)}{\log (dn)},$$ converges or not when $N$ goes to infinity?