## Inequality involving sum of logarithms and hidden zeta-function

I would like to prove the following estimation: if $n \ge 2$ is a natural number, then $$\sum_{k=2}^n \frac{\log^2 k}{k^2} <2 – \frac{\log^2 n}{n}.$$ I have noticed that LHS is indeed bounded by proving that $$\sum_{k = 2}^\infty \frac{\log ^2 k}{k^2} = \zeta”(2) \approx 1.98928$$ and then check with an aid of computer that for … Read more

## Series for ζ(3)−65\zeta(3)-\frac{6}{5}

ζ(2) The inequality 9<π2<10 can be obtained from series ζ(2)=π26=32+12∞∑k=01(k+1)2(k+2)2 and ζ(2)=π26=53−∞∑k=11(k+1)(k+2)2(k+3) Both of them have constant numerator and the denominator a polynomial of fourth degree. ζ(3) Similarly, for Apéry’s constant we can write the inequality 65<ζ(3)<54 from series ζ(3)=65+∞∑k=21k3+4k7 and ζ(3)=54−∞∑k=01(k+1)(k+2)3(k+3) However, the degree of the polynomials is not the same in this case. … Read more

## How does one prove that +∞∑n=11n1+2p≤1+2pp\sum\limits_{n=1}^{+\infty} \frac{1}{n^{1+2p}} \leq \frac{1+2p}{p}?

Got stuck on this while reading a paper. How does one prove that for p>0 the following inequality holds? +∞∑n=11n1+2p≤1+2pp So far I got that the series’ sum is the value of the Riemann ζ-function at z=1+2p. But this gets me no further. Answer Note that since x→1/x1+2p is decreasing in [1,+∞), +∞∑n=11n1+2p=1++∞∑n=21n1+2p≤1++∞∑n=2∫nn−1dxx1+2p=1+∫∞1dxx1+2p=1+12p. AttributionSource : … Read more

## Fourier Transform of the Riemann zeros (Dirac comb)?

Lets assume RH and $\rho_i, i\in\Bbb N$ be the imaginary parts of the non-trivial zeros of the Riemann $\zeta$ function: $\zeta(\frac{1}{2}\pm\imath \rho_i)=0$, $(\forall i)$. Does anonye know if anything (in case what) is known on the (real) Fourier-Transform of a “zeta-zero-Dirac-comb”: $$\mathcal{F}\left \{ \sum_{i=1}^{\infty} \delta(t – \rho_i ) + \delta(t + \rho_i)\right \}[s]$$ … Read more

## Infinite product of Zetafunctions

It is well-known (I learned about this first in a video by Papa Flammy) that ∑∞n=2(ζ(n)−1)=1. This result on its own is quite remarkable, but it also implies convergence of the infinite product P=∞∏n=2ζ(n). I somehow got invested in this. Upon trying to work out the exact value of P, I found that P=∞∑n=1c2(n)n, where … Read more

## A new(?) analytic continuation for the Riemann zeta function.

While tweaking the definition for the Euler gamma constant I found that the following appears to be true: ζ(s)=lim when \Re(s)>0, a>0 and b>0. Can you prove it? Answer Use the AFE (approximate functional equation): \zeta(s)=\sum_{k \le x}k^{-s}-\frac{x^{1-s}}{1-s}+O(x^{-\sigma}) uniform in \sigma \ge \sigma_0>0 and valid for say |t| < \pi x, s=\sigma+it Fixing s, \Re … Read more

## Meaning of equality in zeta regularization

It is known that $$\sum\limits_{n = 1}^\infty{n = 1 + 2 + 3 + \cdots} = \infty$$ but it is also known that $$\sum\limits_{n = 1}^\infty{n = 1 + 2 + 3 + \cdots} = -\frac{1}{{12}}$$ which can obtained using the analytic continuation of $\zeta(s)$. My question is: What is the true meaning of equality … Read more

## How to prove that Riemann zeta function is zero for negative even numbers? [duplicate]

This question already has answers here: How one can obtain roots at the negative even integers of the Zeta function? (2 answers) Closed 6 years ago. Can anyone please explain to me how to prove that Riemann zeta function is 0 for all negative even numbers. In many references , they have just given the … Read more

## Does a generating function for $\zeta(2k+1)$ exist?

I know that a generating function for the Zeta function at the even integers already exists, but how about the Zeta function at the odd integers? I’ve done some research, and found some alternative formulas for the harmonic numbers that allowed me to create a generating function for $\zeta(2k+1)$, but I’d like to know if … Read more

## Non trivial zeros of Riemann zeta function

The non trivial zeros of Riemann zeta function , x\zeta(s) lies in the critical strip 0<\Re(s)<1 Riemann Hypothesis states that all the zeros of Riemann zeta function, \zeta(s) lies on the critical line , \Re(s)=1/2. G.H. Hardy proved that an infinity of zeros are on the critical line, \Re(s)=1/2 Question Are the number of non … Read more