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4 votes
1 answer
213 views

Asymptotic behavior of weighted sums involving the fractional part function

Currently, I am studying the asymptotic behavior of sums of the form \begin{equation}\label{eq1}\tag{1} \sum_{k=1}^{n-1} f(n-k) \left\{ \frac{n}{k} \right\} \end{equation} In this context, based on ...
 Babar's user avatar
  • 611
1 vote
0 answers
113 views

Are there any known statistics on the sign of the Stieltjes Constants?

The Stieltjes Constants $\gamma_n$ arise from considering the laurent series of the Riemann Zeta function at $s=1$ $$ \zeta(s) = \frac{1}{s-1} + \sum_{n=0}^{\infty} (-1)^n \frac{\gamma_n}{n!} (s-1)^n $...
Sidharth Ghoshal's user avatar
1 vote
0 answers
128 views

On Zudilin's linear forms in $1,\zeta(5)$ and $\zeta(7)$

I am reading an article "Well-poised hypergeometric service for diophantine problems of zeta values" by W. Zudilin. Consider the quantities defined here in pg. $617$ $$\tilde{F_n}:= \frac{1}{...
Max's user avatar
  • 11
3 votes
0 answers
167 views

A sharper estimate for a generalization of the sum-of-divisors function

I am interested in the function $f_n(m)$ which can be defined by the Dirichlet generating function $$\zeta(s)\zeta(s - 1) \cdots\zeta(s - n + 1) = \sum\limits_{m = 1}^\infty \frac{f_n(m)}{m^s} $$ This ...
Bear's user avatar
  • 31
5 votes
1 answer
427 views

Lindelöf hypotheses for derivatives of zeta

The Lindelöf hypothesis says that if we have: $$\zeta(\sigma+iT)=\mathcal O(T^a)$$ Then if one considers $\sigma=1/2$ then $\inf a=0$. Further, from convexity and the functional equation this implies ...
psubodiosa's user avatar
7 votes
2 answers
720 views

On the asymptotic behaviour of the series $\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$ near $s=0$

I am interested in determining the behaviour of the the series/function $$f(s)=\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$$ near $s=0$. It is clear that $f(0)$ is undefined....
Tian Vlašić's user avatar
0 votes
0 answers
157 views

Does it make sense to express upper bounds on arithmetic sequences with Dirichlet generating functions?

In order to see what happens when taking the functional equation in this form: $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s) $$ $$\xi(s) = \xi(1 - s)$$ $$\pi^{-s/2}\ \Gamma\left(\...
Mats Granvik's user avatar
  • 1,183
6 votes
1 answer
369 views

Asymptotic estimate for an integral involving the squared modulus of the Riemann zeta function

For any fixed $\frac{1}{2} < \sigma < 1$, let $$\int_0^T \frac{|\zeta(\sigma+it)|^2}{\sqrt{1+t^2}} \ dt = O(T^\theta), \qquad T \uparrow \infty. $$ It is clear that $\theta > 0$, since we ...
nickkatzfl's user avatar
2 votes
1 answer
197 views

Interpretation of an equivalence to the Riemann hypothesis due to de Reyna and Toulisse in the spirit of a formula from an article

In [1] the authors present an equivalence to the Riemann hypothesis that is the Theorem 6.2. On the other hand I know a statement from [2], in English this is the article Andrew Granville and Greg ...
user142929's user avatar
0 votes
2 answers
339 views

Error term in França-LeClair approximation of zeta zeros

The imaginary part of the $n$th critical zero of the Riemann zeta function with positive imaginary part (in increasing order) is asymptotically $$ t_n \sim 2\pi\frac{n}{\log n} $$ and has been ...
Charles's user avatar
  • 9,114
6 votes
1 answer
4k views

About the logarithmic derivative of the Riemann zeta function

Let $\rho=\beta+i\gamma$ a non-trivial zeros of the Riemann zeta function and $s=\sigma+it$ a complex number. It is possible to prove that $$\frac{\zeta'}{\zeta}\left(s\right)=\sum_{\left|t-\gamma\...
User's user avatar
  • 219