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Δημοσιεύτηκε: **Πέμ Σεπ 10, 2026 10:16 pm**

Να αποδείξετε ότι για κάθε ζεύγος (m,n)

θετικών ακεραίων με

n\equiv 1\left (\mod4\right ),

ισχύει

n!\left ( \sum_{\theta =0}^{\infty} e^{-\left ( 2\theta +1 \right )m}\sum_{0\leq p\leq \left \lfloor \dfrac{n}{4} \right \rfloor } \frac{\pi ^{4p}\zeta \left ( n-4p+1 \right )}{\left ( 4p \right )!\left ( 2\theta +1 \right )^{n-4p+1}}+\sum_{q=1}^{\infty}e^{-2qm}\sum_{0\leq i\leq \left \lfloor \dfrac{n-2}{4} \right \rfloor} \frac{\pi ^{4i+2}\zeta \left ( n-4i-1 \right )}{\left ( 4i+2 \right )!\left ( 2q \right )^{n-4i-1}}\right )-\left ( \sum_{\theta =0}^{\infty}e^{-\left ( 2\theta +1 \right )m} \sum_{0\leq i\leq \left \lfloor \dfrac{n-2}{4}\right \rfloor} \binom{n}{4i+2}\pi^{ 4i+2}\left ( \int \limits_{0}^{\infty}x^{n-4i-2}\left ( \frac{1}{e^{x}-1} \right )^{2\theta +1}dx \right )+\sum_{q=1}^{\infty}e^{-2qm}\sum_{0\leq p\leq \left \lfloor \dfrac{n}{4}\right \rfloor}\binom{n}{4p}\pi ^{4p}\left ( \int \limits_{0}^{\infty}x^{n-4p}\left ( \frac{1}{e^x-1} \right )^{2q}dx \right )\right ) \leq \frac{n!\zeta \left ( n+1 \right )}{m^{n+1}}+\frac{\pi ^{n+1}}{2\left ( n+1 \right )}

Ιδιοκατασκευή Ανισότητα!