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Your data is compatible with the more refined estimates proved by Vaughan in Ramanujan J. 15 (2008), 109–121. His Theorems 1 and 2 (together with his (1.9)) reveal that $$\log(A000607(n)) = 2 \pi \sq …

See Erick Wong's response here. In particular, Kevin Ford proved (in more precise form) that $$ f(n) = \frac{n}{\log n} \exp\left(O(\log \log \log n)^2\right),$$ whence $f(n)/n$ tends to zero. The sam …

The asymptotic formula is true for even dimensions $k\geq 2$. We can prove this by induction on $k$, inspired by Rodrigo's observation on Eisenstein series. The case $k=2$ is classical and addressed …

I think your final goal follows by taking the logarithmic derivative of the functional equation: $$\frac{\zeta'}{\zeta}(s)+\frac{\zeta'}{\zeta}(1-s)=\log\pi-\frac{1}{2}\frac{\Gamma'}{\Gamma}\left(\fra …

Let us denote the left hand side of $(1)$ by $\psi(x)$. It is known that $|\psi(x)-x|$ is not bounded by a constant times $x^{1/2}$. In fact Littlewood (1914) proved that $$\psi(x)-x=\Omega_{\pm}(x^{1 …

The Lindelöf hypothesis yields the same bound for each derivative of $\zeta(s)$ via Cauchy's formula. Indeed, let $n\in\mathbb{N}$, $\sigma\in\mathbb{R}$, $T\in(1,\infty)$, $\varepsilon\in(0,1/2)$. Le …

You can find an in-depth answer to your question in this paper of de Reyna and Jeremy. See in particular (65)-(66) along with (30) and Theorem 4.9. See also Theorem 6.2.

The Riemann hypothesis is equivalent to the following statement: $$f(x)=\mathrm{li(x)}-\frac{x}{\log x}+O(\sqrt{x}),\qquad x\geq 2.$$ Note that $$\mathrm{li(x)}=\mathrm{li(2)}+\frac{x}{\log x}-\frac{2 …

Your sum can be rewritten as $$ D(N)=\sum_{n=1}^{N-1}d(n)d(N-n),$$ where $N=2c$, and $d(m)$ is the number of divisors of $m$. This is a so-called "binary additive divisor sum", and it has been studied …

Yes, these are standard things in analytic number theory. We have $$ \sum_{n=1}^\infty\frac{\tau(n)^k}{n^s}=\zeta(s)^{2^k}F_k(s),\qquad\Re(s)>1,$$ where $F_k(s)$ is an explicit Dirichlet series that c …

You can derive a very precise asymptotic expansion of your quantity by the Selberg-Delange method. I recommend that you adapt, to your situation, the arguments of Section II.6.1 of Tenenbaum: Introd …

This is my third response. I claim that for $0 < t < 1$ we have the uniform bounds $$ \liminf_{n\to\infty}\ nt f_n(t) = \frac{-1}{\log\mu}+O(1),$$ $$ \limsup_{n\to\infty}\ nt f_n(t) = \frac{-1}{\log\m …

Let us introduce the notation $$M(T):=\int_0^T|\zeta(\sigma+it)|^2\,dt.$$ Then $$\int_0^T \frac{|\zeta(\sigma+it)|^2}{\sqrt{1+t^2}} \,dt=\int_0^T\frac{dM(t)}{\sqrt{1+t^2}}=\frac{M(T)}{\sqrt{1+T^2}}+\i …

The sum in question equals \begin{align*}\sum_{p\leq x}\frac{1}{p^{1+2/\log x}}\left(\frac{\log\left(x/p\right)}{\log(x)}\right)^2 &=\sum_{p\leq x}\frac{1}{p}e^{-2\frac{\log p}{\log x}}\left(1-\frac{\ …

$F(H)$ and $G(H)$ do not determine the asymptotic growth of the third display. Indeed, consider the following two functions from $\mathbb{Z}_{\geq 0}$ to $\mathbb{Z}_{\geq 0}$: $$ f(x):=\begin{cases}0 …