# Tagged Questions

Questions on the famed $\zeta(s)$ function of Riemann, and its properties.

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### An interesting identity involving powers of $\pi$ and values of $\eta(s)$

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
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### A question about theorem 2 in de Bruijn's 1950 paper “The roots of trigonometric integrals”

Theorem 2 of de Bruijn's paper titled "The roots of trigonometric integrals" (Duke Math. J., 17 (1950)) is given by: What does it mean by "the function $q(x)$ be regular in the sector...? Does it ...
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the Euler product for the Riemann zeta $$\zeta (s)= \prod _{p}\left( \sum_{n=0}^{\infty}p^{-ns}\right)$$ this is only valid for $\Re(s) >1$ however we could use the Borel transform so $$\... 1answer 56 views ### Analytic continuation of Zeta type function Can one analytically continue the function (Not equal to the Zeta function)$$Z(s)=\prod_{p}\frac{1}{1+p^{-s}}=\sum_{k=1}^{\infty}\frac{(-1)^{\Omega(k)}}{k^s}$$Where \Omega(k) is the number of ... 1answer 267 views ### An integral that might be related to the modified Bessel function of second kind It is known that the modified Bessel Function K_z(a) (a>0)can be expressed as a Fourier transform$$K_z(a)=\frac{1}{2}\int_{-\infty}^{\infty}\exp(-a\cosh t)\cosh(zt){\rm d}t=K_{-z}(a)$$Can ... 2answers 61 views ### Zeta Function \zeta(1\pm1/n) and Euler's constant. How do I show that$$\lim_{n\to\infty}{\zeta(1+1/n)+\zeta(1-1/n)}=2\gamma$$and$$\lim_{n\to\infty}{\zeta(1+1/n)-\zeta(1-1/n)}=\infty,$$where \gamma is the Euler's constant? 2answers 589 views ### Simpler zeta zeros Is it true that$$\lim_{y\rightarrow\infty}\dfrac{\sum_{n=1}^{y}n^{-1/2-iy}}{\zeta(1/2+iy)}=1$$? Below is a plot of$$\sum_{n=1}^{y}\dfrac{1}{n^{s}}\text{for }s=\dfrac{1}{2}+iy$$set against its ... 1answer 129 views ### How to show \sum_{n=1}^{\infty}\frac{H_{n}}{n^{2}}=2\zeta (3)? [duplicate] How to show this equation is true.$$\sum_{n=1}^{\infty}\frac{H_{n}}{n^{2}}=2\zeta (3)$$where H_{n}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n} 1answer 50 views ### Why does the imag. part of the graph of \zeta(n^{ix}) resemble the tangent function? If you input \zeta(n^{ix}) into the Wolfram Alpha search bar, in the plot, you get an infinitely repeating sinusoidal curve, which resembles the real part, and you get an infinitely repeating ... 0answers 107 views ### Prime Zeta Function Does$$\sum_{p \text{ prime}} \frac{1}{p^s} \sim \log \zeta(s) \quad \text{as} \quad s \to 1^+$$imply$$\sum_{p \leq n} \frac{1}{p} \sim \log H_n \quad \text{as} \quad n \to \infty,$$where H_n is ... 0answers 27 views ### Explicit formulas for Fourier coefficients from its Tayor expansion In my research, I need to determine unique coefficients a_k in terms b_k:$$\sum_{k=0}^n a_k \cos\left(\frac{k}{n+1}t\right)+O\left(t^{2n+1}\right)=\sum_{k=0}^n b_k t^{2k}.$$This problem showed ... 4answers 135 views ### fractional part of Riemann zeta \sum_{s=2}^\infty \{\zeta (s)\}=1$$\sum_{s=2}^\infty \{\zeta (s)\}=1 where $\zeta (s)$ is Riemann zeta, $\{x\}$ denotes the fractional part of the real number $x$ The problem was proposed by Michael Th. Rassias $\{\zeta(2)\}=\... 0answers 50 views ### How to evaluate Bessel functions$K_{i x+\frac{9}{4}}(2\pi)+K_{i x-\frac{9}{4}}(2\pi)$with$x>1.5*10^8$in Mathematica 7? I am trying to evaluate Bessel functions$K_{i x+\frac{9}{4}}(2\pi)+K_{i x-\frac{9}{4}}(2\pi)$with$x>1.5*10^8$in Mathematica 7. This function is the first Polya approximation to Riemann$\xi(s)$... 0answers 85 views ### Investigating the convergence of a series using the comparison limit test, Part II [duplicate] I posted this question earlier, but as I don't know if a comment reply or edit will refresh this so people actually see, I'm going to post it again in hopes that someone knows what's going on. Here's ... 0answers 91 views ### English translation of two papers by Polya on real zeros of Fourier transform approximation to Riemann$\xi$function I am looking for English translation of the following two papers by Polya: [1] G. Polya, Bemerkung über die Integraldarstellung der Riemannschen$\xi$-Funktion, Acta Math. 48(1926), 305-317; ... 1answer 76 views ### Product of zeta and its conjugate Suppose we have the zeta function$\zeta(s)$, and we want to multiply it by its complex conjugate$\zeta(s)^*$. Since$\zeta(s)^* = \zeta(s^*)$, we get$\displaystyle \zeta(s)\cdot\zeta(s)^* = \left[...
I'm probably being really stupid but in a proof of the Laurent expansion of the Riemann zeta function the quantity $$S_r(t) = \sum_{n \leq t} \frac{(\log (x/n))^r}{n}$$ is ...
### Calculating the residues of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$
Calculating the poles of $\frac{\zeta^{\prime}{(s) x^{s}}}{\zeta(s)\cdot s}$, where x is a fixed real number. I am trying to calculate the poles of this function at the trivial zeros of $\zeta$, ...