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Mathematics
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Ali
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7
How to bound $\mu (\{ x \le X : |ψ(x) − x| \ge εx^{1/2} (\log x)^2 \})$ from above?
Apr 27
6
Show that $\dfrac{1}{e^x-1} = \dfrac{1}{x} - \dfrac{1}{2} + 2x \sum_{n=1}^{\infty} \dfrac{1}{(4n^2 \pi^2 +x^2)}$.
Jan 8
6
Show that $\zeta(s) = - \dfrac{\sin \pi s}{\pi} \int_0^{\infty} \bigg( \dfrac{\Gamma'(1+x)}{\Gamma(1+x)} - \log x \bigg) x^{-s} dx$.
Jan 18
4
Show that $\dfrac{1}{2 \pi i} \int_{c- i \infty}^{c+ i \infty} F(s) G(w-s) ds = \int_0^{\infty} f(x) g(x) x^{w-1} dx.$
Jan 10
4
What does "behaving like independent random variables" mean?
May 11
3
Show that $\dfrac{1}{4} \sum_{n=1}^{\infty} \dfrac{r(n)}{n^w} - \zeta(2w) = \zeta(w) \eta(w) - \zeta(2w)$.
Jan 11
3
Understanding Lemma 4.6. of Titchmarsh's book
Jan 17
3
Understanding the proof of Theorem 10.1 in Montgomery & Vaughan's Multiplicative Number Theory
Feb 26
2
Why $L(s, χ)$ is nonzero for $s$ real and $χ$ complex?
May 18
2
On the 'reverse theorem' for functional equation of the Riemann zeta function
Jan 11