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Assume the Riemann hypothesis. We know that $$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| \leq 2 \log \log t + O(1)$$ (see, e.g., Thm. 13.13 in Montgomery-Vaughan). What is the best explicit bound o …

Let $\mu$ be the Möbius function. Let $$\lambda_d = \begin{cases} \frac{\log D/d}{\log D} \mu(d)&\text{for $d\leq D$,}\\ 0 &\text{otherwise.}\end{cases}$$ (Selberg's weights also work.) Then it is wel …

Let $$F(x) = \sum_{d\leq x} \frac{\mu(d)}{d} \log \frac{x}{d}.$$ s it possible/feasible to give an elementary proof of the fact that $F(x)= 1 + o(1)$ (and, ideally, $1+O(1/\log x)$, or better)? By " …

For $x\geq 1$, $$\rho(x) = \sum_{n\leq x} \frac{\mu(n)}{\sigma(n)} \log \frac{x}{n}.$$ As $x\to\infty$, this sum tends to $\zeta(2) = \pi^2/6$. Is it in fact the case that $\rho(x)\leq \zeta(2)$ for …

I was leafing through Gradshteyn–Ryzhik in bed yesterday, as one does, and noticed on the last page that the Mellin transforms of several hyperbolic functions have a factor of $\zeta(s-1)$ or $\zeta(s …

Let $Y$ be the set of integers $N<n\leq 2 N$ with more than $D \log \log N$ prime factors. We may consider, say, $D = (\log \log N)^{1-\epsilon}$. We do have rather precise approximations for the siz …

Are there effective, explicit upper bounds on $L'(1,\chi)/L(1,\chi)$ in the literature, valid for all Dirichlet characters $\chi$? Due to the possibility of Siegel zeros, I don't imagine one can do mu …

Let $$I(T)=\int_0^T \left|\zeta\left(\frac{1}{2} + i t\right)\right|^2 dt$$ and let $E(T)$ be $I(T)$ minus what turn out to be its main terms: $$E(T) = I(T)- T \log \frac{T}{2 \pi} - (2 \gamma - 1) T. …

The following approach to the divisor problem (that is, the problem of estimating $D(x) = \sum_{n\leq x} d(n)$, where $d(n)$ is the number of divisors of $n$; more precisely, we are meant to bound the …

What are some different ways to achieve the bound $$\int_0^T \left|\zeta\left(\frac{1}{2} + i t\right)\right|^2 dt = T \log \frac{T}{2 \pi} + (2 \gamma - 1) T + E(T)$$ with an error term $E(T) = O(T^{ …

Say that an integer $n$ is $p$-leading if its expansion in base $p$ starts with the digit $p-1$. My postdoc, Lifan Guan, asks: are there infinitely many positive integers $n$ that are not $p$-leading …

Are there bounds in the literature on sums of the form $$\sum_{\omega(n)= k} e(\alpha n) \;\;\;\;\;\text{or}\;\;\;\;\; \sum_{\Omega(n)=k} e(\alpha n)$$ for $\alpha$ on minor arcs (i.e., not very close …

What sort of bounds (explicit of preference) can one give for $$\int_T^{2 T} \frac{dt}{|\zeta(1+i t)|^2} \;\;\;\;\;?$$ Some obvious points: One can give a pointwise bound $\frac{1}{|\zeta(1+ it)|} \l …

Is there an explicit estimate in the literature bounding from above the number of square-free numbers in a short interval $x<n\leq x y$? I can easily do this by means of the Selberg sieve, but I do no …

Let $y>x\geq 1$, $p_0\geq x$. Consider $$S=\mathop{\sum_{x\leq p\leq y}}_{p\equiv a \mod p_0} \frac{1}{p}.$$By Brun-Titchmarsh and (basically) integration by parts, I seem to get that $$S \leq \frac{1 …