Question Number 141082 by mnjuly1970 last updated on 15/May/21
$$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:……..\:{nice}\:\:…….\:\:{calculus}\:…….. \\ $$$$\:\:\:\boldsymbol{\phi}:=\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\:\:\frac{\zeta\:\left(\:{n}\:\right)}{{n}\:.\:\mathrm{4}^{{n}} }=? \\ $$$$ \\ $$
Answered by mindispower last updated on 15/May/21
$$\Gamma\left({x}+\mathrm{1}\right)\zeta\left({x}+\mathrm{1}\right)=\int_{\mathrm{0}} ^{\infty} \frac{{t}^{{x}} }{{e}^{{t}} β\mathrm{1}}{dt} \\ $$$$\phi=\underset{{n}\geqslant\mathrm{2}} {\sum}\frac{\mathrm{1}}{\Gamma\left({n}\right)}.\frac{\mathrm{1}}{{n}\mathrm{4}^{{n}} }\int_{\mathrm{0}} ^{\infty} \frac{{t}^{{n}β\mathrm{1}} }{{e}^{{t}} β\mathrm{1}}{dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\underset{{n}\geqslant\mathrm{2}} {\sum}.\frac{\mathrm{1}}{\Gamma\left({n}+\mathrm{1}\right)}\int_{\mathrm{0}} ^{\infty} \left(\frac{{t}}{\mathrm{4}}\right)^{{n}β\mathrm{1}} .\frac{\mathrm{1}}{\boldsymbol{{e}}^{\boldsymbol{{t}}} β\mathrm{1}}\boldsymbol{{dt}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\int_{\mathrm{0}} ^{\infty} .\frac{\mathrm{4}}{{t}}.\underset{{n}\geqslant\mathrm{2}} {\sum}\left(\frac{\left(\frac{{t}}{\mathrm{4}}\right)^{{n}} }{{n}!}\right).\frac{\mathrm{1}}{{e}^{{t}} β\mathrm{1}}{dt} \\ $$$$=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{{t}}\left(\frac{{e}^{\frac{{t}}{\mathrm{4}}} β\mathrm{1}β\frac{{t}}{\mathrm{4}}}{{e}^{{t}} β\mathrm{1}}\right){dt} \\ $$$${let}\:{f}\left(\alpha\right)=\int_{\mathrm{0}} ^{\infty} \frac{{e}^{\alpha{t}} β\mathrm{1}β\alpha{t}}{{t}\left({e}^{{t}} β\mathrm{1}\right)}{dt} \\ $$$$\phi={f}\left(\frac{\mathrm{1}}{\mathrm{4}}\right) \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$${f}'\left(\alpha\right)=\int_{\mathrm{0}} ^{\infty} \frac{{te}^{\alpha{t}} β{t}}{{t}\left({e}^{{t}} β\mathrm{1}\right)}{dt}=\int_{\mathrm{0}} ^{\infty} \frac{{e}^{\alpha{t}} β\mathrm{1}}{{e}^{{t}} β\mathrm{1}}{dt},\mathrm{0}\leqslant\alpha<\mathrm{1} \\ $$$${f}'\left({a}\right)=\int_{\mathrm{1}} ^{\infty} \frac{{t}^{{a}} β\mathrm{1}}{{t}β\mathrm{1}}.\frac{{dt}}{{t}} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\frac{\mathrm{1}}{{t}^{{a}} }β\mathrm{1}}{\mathrm{1}β{t}}.{dt} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{t}^{β{a}} β\mathrm{1}}{\mathrm{1}β{t}}{dt}=β\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}β{t}^{β{a}} }{\mathrm{1}β{t}}{dt},\Psi\left({x}+\mathrm{1}\right)=β\gamma+\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}β{t}^{{x}} }{\mathrm{1}β{t}} \\ $$$${f}'\left({a}\right)=β\Psi\left(\mathrm{1}β{a}\right)β\gamma \\ $$$${f}\left({a}\right)={log}\left(\Gamma\left(\mathrm{1}β{a}\right)\right)β\gamma{a}+{c} \\ $$$${f}\left(\mathrm{0}\right)={c}=\mathrm{0} \\ $$$$\phi={f}\left(\frac{\mathrm{1}}{\mathrm{4}}\right)={log}\left(\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\right)β\frac{\gamma}{\mathrm{4}} \\ $$$$ \\ $$
Commented by mnjuly1970 last updated on 15/May/21
$${bravo}\:{bravo}\:{sir}\:{power}… \\ $$
Commented by mindispower last updated on 15/May/21
$${withe}\:{pleasur}\:{Sir} \\ $$
Answered by mnjuly1970 last updated on 15/May/21
$$\:\:\:\:\:\:\Gamma\left({x}+\mathrm{1}\right):=\:{e}^{β\gamma{x}} \underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\frac{{e}^{\frac{{x}}{{n}}} }{\mathrm{1}+\frac{{x}}{{n}}} \\ $$$$\:\:\:\:\:\:\:\:\:{ln}\left(\Gamma\left({x}+\mathrm{1}\right)\right):=β\gamma{x}+\underset{{n}\geqslant\mathrm{1}} {\sum}\left(\frac{{x}}{{n}}β{ln}\left(\mathrm{1}+\frac{{x}}{{n}}\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:\frac{{ln}\left(\Gamma\left({x}+\mathrm{1}\right)\right)}{{x}}:=β\gamma\:+\underset{{n}\geqslant\mathrm{1}} {\sum}\left(\frac{\mathrm{1}}{{n}}β\frac{\mathrm{1}}{{x}}\underset{{k}=\mathrm{1}} {\sum}\frac{\left(β\mathrm{1}\right)^{{k}β\mathrm{1}} {x}^{{k}} }{{k}.{n}^{{k}} }\right) \\ $$$$\:\:\::=β\gamma+\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{1}}{{n}}β\underset{{n}\geqslant\mathrm{1}} {\sum}\underset{{k}\geqslant\mathrm{1}} {\sum}\frac{\left(β\mathrm{1}\right)^{{k}β\mathrm{1}} {x}^{{k}β\mathrm{1}} }{{k}.{n}^{{k}} } \\ $$$$\:\::=β\gamma+\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{1}}{{n}}β\underset{{k}\geqslant\mathrm{1}} {\sum}\left(\frac{\left(β\mathrm{1}\right)^{{k}β\mathrm{1}} {x}^{{k}β\mathrm{1}} }{{k}}\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{1}}{{n}^{{k}} }\right) \\ $$$$\:\::=β\gamma+\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{1}}{{n}}β\underset{{k}\geqslant\mathrm{1}} {\sum}\frac{\left(β\mathrm{1}\right)^{{k}β\mathrm{1}} {x}^{{k}β\mathrm{1}} }{{k}}\zeta\left({k}\right) \\ $$$$\:=β\gamma\:+\zeta\left(\mathrm{1}\right)β\zeta\left(\mathrm{1}\right)+\frac{\zeta\left(\mathrm{2}\right){x}}{\mathrm{2}}β\frac{\zeta\left(\mathrm{3}\right){x}^{\mathrm{2}} }{\mathrm{3}}+… \\ $$$$\:=β\gamma+\underset{{k}\geqslant\mathrm{2}} {\sum}\frac{\left(β\mathrm{1}\right)^{{k}} {x}^{{k}β\mathrm{1}} \zeta\left({k}\right)}{{k}} \\ $$$$\:\frac{{ln}\left(\Gamma\left({x}+\mathrm{1}\right)\right)}{{x}}=β\gamma+\frac{\mathrm{1}}{{x}}\underset{{k}\geqslant\mathrm{2}} {\sum}\frac{\left(β\mathrm{1}\right)^{{k}} {x}^{{k}} \zeta\left({k}\right)}{{k}} \\ $$$$\:\:\:{ln}\left(\Gamma\left({x}+\mathrm{1}\right)\right)=β\gamma{x}+\underset{{k}\geqslant\mathrm{2}} {\sum}\frac{\left(β\mathrm{1}\right)^{{k}} {x}^{{k}} \zeta\left({k}\right)}{{k}} \\ $$$$\:\:\:{x}:=β\frac{\mathrm{1}}{\mathrm{4}}\:\Rightarrow\:{ln}\left(\Gamma\left(\mathrm{1}β\frac{\mathrm{1}}{\mathrm{4}}\right)\right)=\frac{\gamma}{\mathrm{4}}+\underset{{k}\geqslant\mathrm{2}} {\sum}\:\left(\frac{\zeta\left({k}\right)}{{k}.\mathrm{4}^{{k}} }\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{ln}\left(\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\right)β\frac{\gamma}{\mathrm{4}}=\underset{{k}\geqslant\mathrm{2}} {\sum}\:\left(\frac{\zeta\:\left({k}\:\right)}{{k}\:.\:\mathrm{4}^{\:{k}} }\:\right) \\ $$
Commented by mindispower last updated on 15/May/21
$${sir}\:{are}\:{you}\:{student}\:? \\ $$
Answered by Dwaipayan Shikari last updated on 15/May/21
$$\psi\left({z}+\mathrm{1}\right)=β\gamma+\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\zeta\left({n}\right){z}^{{n}β\mathrm{1}} \\ $$$$\left[{log}\left(\Gamma\left({z}+\mathrm{1}\right)\right)\right]_{\mathrm{0}} ^{β\frac{\mathrm{1}}{\mathrm{4}}} =β\int_{\mathrm{0}} ^{\frac{β\mathrm{1}}{\mathrm{4}}} \gamma+\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\int_{\mathrm{0}} ^{β\frac{\mathrm{1}}{\mathrm{4}}} \left(β\mathrm{1}\right)^{{n}} \zeta\left({n}\right){z}^{{n}β\mathrm{1}} {dz} \\ $$$${log}\left(\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\right)=\frac{\gamma}{\mathrm{4}}+\Sigma\frac{\zeta\left({n}\right)}{{n}\mathrm{4}^{{n}} } \\ $$$$\Sigma\frac{\zeta\left({n}\right)}{{n}\mathrm{4}^{{n}} }={log}\left(\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\right)β\frac{\gamma}{\mathrm{4}} \\ $$
Commented by mnjuly1970 last updated on 15/May/21
$$\:{very}\:{very}\:{nice} \\ $$$${thank}\:{you}\:{mr}\:{payan}… \\ $$$$\:{taylor}\:{expansion}\:{of}\:{digamma}.. \\ $$