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Implications of divergence of $1/\zeta(s) $ at 1/2

$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function. This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$. Its convergence is unknown if $1/2< s&...
Koushik's user avatar
  • 2,106
7 votes
0 answers
179 views

When does the function $F(x)=\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$ reach $F(x) > 8$?

We know from Ramanujan and Riemann that, $$\pi(x) = \operatorname{li}(x) -\tfrac12\operatorname{li}(x^{1/2})-\tfrac13\operatorname{li}(x^{1/3})-\tfrac15\operatorname{li}(x^{1/5}) +\dots$$ with prime ...
Tito Piezas III's user avatar
5 votes
0 answers
137 views

Is finding positive integer solutions of $\zeta(a/b) = c$ equivalent to deciding the rationality of $\gamma$?

This question requires little bit of explanation of the background hence it is a bit lengthy. Note: The question was initially posted in MSE but did not get answers hence posting in MO. For every ...
Nilotpal Kanti Sinha's user avatar
5 votes
0 answers
287 views

Are there infinitely many zeros of $\chi(s)+ \dfrac{2^{s}- 2^{2s-1}}{2^{s}-1}$ on the critical line?

Take $\chi(s)= 2^s\,\pi^{s-1}\,\sin\left(\frac{\pi\,s}{2}\right)\,\Gamma(1-s)$, so that $\zeta(s)=\chi(s)\,\zeta(1-s)$. The zeros of $\chi(s)=-1$ and the non-trivial zeros $\rho$ of $\zeta(s)$, seem ...
Agno's user avatar
  • 4,169
3 votes
0 answers
171 views

Estimating integral of product of terms $\cos(t\log p)$

I would like to prove the following proposition from A. Harper's paper "Sharp conditional upper bound for moments of the Riemann Zeta Function" Proposition. Let $T$ be large and let $n=p_1^{\...
asd's user avatar
  • 199
3 votes
0 answers
139 views

Square integral of finite Euler product

Consider the finite Euler product $$ P(t) = \prod_{r=1}^R \left(1 + p_r^{i t} \right). $$ (Here $p_1, p_2, \dots$ are of course the primes.) Question: What is a good asymptotic upper bound for $$ \...
Kurisuto Asutora's user avatar
2 votes
0 answers
238 views

Possible regularisation for sum of function of primes

Consider the following sum of function of primes: $$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$ Here $p$ runs through all primes and $e$ is Euler's constant. We can see that the sum ...
Zaza's user avatar
  • 149
2 votes
0 answers
157 views

How could this difference in series of power of zeros associated to counting integers and counting primes be explained?

Introduction: In this 1992 paper, J.B. Keiper (an amazing person, who tragically died way too young), derives several power series expansions of the Riemann $\xi$-function that involve infinite sums ...
Agno's user avatar
  • 4,169
2 votes
0 answers
313 views

Proving that the Riemann zeta function is zero free on Re=1 using the prime number theorem

Is $\frac{-\zeta'(s)}{\zeta(s)}+\frac{-s}{s-1}$ an analytic continuation, holomorphic for $Re\ s > 0,\ s\neq 1$, of $f(s)=s\int_{1}^{\infty}\frac{\psi(x)-x}{x^{s+1}}\mathrm{d}x$? If so: Let $s_{0}$ ...
Juu's user avatar
  • 129
2 votes
0 answers
537 views

Explicit formula for $n$th prime in terms of Riemann zeros:

We all know there exists an explicit Formula for prime counting function in terms of Riemann zeros. I'm wondering if similar formula exists for $n$th prime in terms of Riemann zeros? Or any other ...
bambi's user avatar
  • 375
2 votes
0 answers
147 views

Skewes' number and the ratio $\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$

(A complementary post is here.) Given the prime counting function $\pi(x)$ and the logarithmic integral $\operatorname{li}(x)$, we have Table 1, $$\begin{array}{|c|l|} \hline x&\operatorname{li}...
Tito Piezas III's user avatar
1 vote
0 answers
155 views

Function involving argument of the Riemann zeta function

When $t$ is an ordinate of a zero of Riemann zeta function, we define \begin{equation} f(t):=\frac{t}{2\pi}\log\left(\frac{t}{2\pi e}\right)+S(t)-\frac{1}{8}+\frac{1}{48 \pi t}+\frac{7}{5760 t^3}+...
Steve's user avatar
  • 19
1 vote
0 answers
482 views

Explicit formula for zeta function with special type of weight

Consider the following line of thinking: $$\pi(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x} $$ Here, $\operatorname{R}(x) = \...
TPC's user avatar
  • 790
1 vote
0 answers
188 views

Questions on Riemann's explicit formula

If we consider this version of the prime-counting function $$\pi_0(x) = \frac{1}{2} \lim_{h\to 0} (\pi(x+h) + \pi(x-h))$$ (with $\pi$ being the normal prime-counting function), then we can write $\...
tobias's user avatar
  • 749
0 votes
0 answers
352 views

On a Duality between Riemann-weil explicit formula and Abel- Plana summation of trigonometric prime counting function:

Consider the analytic function $g(x)$ Now define $f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$ Such that $|f(x+it)|=o(e^{2πt})$ uniformly for every $x$...
TPC's user avatar
  • 790
0 votes
0 answers
169 views

On $\sum_{\rho\in D} \text{dist}(\rho)=\frac{1}{2\pi i} \int_{\partial{D}}\log \zeta(s)\ ds$

Let $D$ denote a closed two dimensional figure as: $D=2+iT\to 2\to 2-\delta\to 2-\delta+i(T-\delta)\to \frac{1}{2}+\epsilon+i(T-\delta)\to\frac{1}{2}+\epsilon\to\frac{1}{2}-\epsilon\to \frac{1}{2}-\...
Honor's user avatar
  • 11
-3 votes
0 answers
71 views

Is the upper bound on $H_{1}$ a decreasing function of the proportion of critical zeros of Zeta?

This question stems from https://arxiv.org/abs/2411.19762 and the numerical observation that the best unconditional upper bound for $H_{1}:=\lim\inf_{n\to\infty}p_{n+1}-p_{n}$, namely $H_{1}^{\flat}=...
Sylvain JULIEN's user avatar