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5 votes
Accepted

Sign of $\Re\frac{\xi'(s)}{\xi(s)}$ locally around a zeta zero

If a complex function $f(s)$ has a zero at $s_0$ of order $n\geq 1$, then for suitable $r>0$, $$\frac{f'(s)}{f(s)}=\frac{n}{s-s_0}+O(1),\qquad 0<|s-s_0|<r.$$ Writing $s=s_0+\rho e^{it}$ with $...
GH from MO's user avatar
  • 101k
5 votes

Least number coprime to a given integer

If $f(n) = p_{m+1}$ is the $(m+1)$-th prime, then $n$ must be divisible by $p_1 \cdots p_m$, hence $f(n) \ll m \log m \ll \omega(n) \log \omega(n)$. Your example shows one cannot hope for a better ...
François Brunault's user avatar
5 votes
Accepted

Is "self-dual" equivalent to "dihedral" for Maass forms on $\mathrm{GL}(2)$?

Let $f$ be a Maass form on the upper half-plane with nebentypus $\chi$. It is known that \begin{align*} \text{$f$ is self-dual}&\qquad\Longleftrightarrow\qquad\text{$L(f\times f,s)$ has a pole at $...
GH from MO's user avatar
  • 101k
4 votes

Density of numbers with a large prime factor in specified arithmetic progression

The same second moment argument, the one that shows almost all $n$ have $\bigl(1+o(1)\bigr)\log\log n$ distinct prime factors, shows that almost all $n$ have $\bigl(\frac{1}{\phi(b)}+o(1)\bigr)\log\...
mathworker21's user avatar
  • 1,078
4 votes
Accepted

On $\Re\frac{\zeta'}{\zeta}(s) \geq -A \log(|t|+4)$ for some $A>0$

If the Riemann Hypothesis fails, then there is no such lower bound. Indeed, if $\zeta(s_0)=0$ and $\Re(s_0)>1/2$, then $$\lim_{\delta\to 0+}\Re\frac{\zeta'(s_0-\delta)}{\zeta(s_0-\delta)}=-\infty.$$...
GH from MO's user avatar
  • 101k
3 votes
Accepted

Estimates for the first coefficient of the Taylor expansion of $\zeta$ around a zero

Since $$c_1=\zeta'(s_0)\qquad\text{and}\qquad c_1'=\zeta'(1-\overline{s_0})=\overline{\zeta'(1-s_0)},$$ the question concerns $|\zeta'(s_0)/\zeta'(1-s_0)|$. The functional equation for $\zeta(s)$ can ...
GH from MO's user avatar
  • 101k
3 votes

Exponential sums involving smooth truncated divisor functions

This should follow by approximating $a/p$ by $a/q+\beta$ with $q^2\ll MN$ and $q|\beta|\ll 1/\sqrt{MN}$. For simplicity, I’ll suppose $\beta=0$ (this will be the Archimedean analogue of the $q$-...
Mayank Pandey's user avatar
3 votes

The lower bound for the automorphic $L$-function $L(s,\pi)$ at the edge of the critical strip $\Re s=1$

This is not known. The best known lower bound is due to Brumley (2013), to be found in an Appendix to a paper by Lapid, which gives that $$|L(1+it,\pi)| \gg_{\pi_{\infty},\varepsilon} ( N (1+|t|) ...
GH from MO's user avatar
  • 101k
2 votes

Comparing sizes of sets of natural numbers

Well, lots of time have passed and now I have an explicit formula for numerosity. In case of uniform lattices it gives the same differences between numerosities of sets as your formula, but can ...
Anixx's user avatar
  • 9,703
2 votes
Accepted

Riemann xi function strictly increasing along a half-plane

The result you mention is not due to Matiyasevich-Saidak-Zvengrowsk. Instead, it appeared in Sondow-Dumitrescu: A monotonicity property of Riemann's xi function and a reformulation of the Riemann ...
GH from MO's user avatar
  • 101k
1 vote

Consequences of Goldbach's conjecture

If $R(2m)$ is the number of Goldbach partitions of 2m and $\pi (m, 2m)$ is the number of primes between m and 2m then $\pi (m , 2m) \geq R(2m) \geq 1$. Bertrand's postulate asserts that the number of ...

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