Questions tagged [zeta-functions]

Questions on various generalizations of the Riemann zeta function (e.g. Dedekind zeta, Hasse–Weil zeta, L-functions, multiple zeta). Consider using the tag (riemann-zeta) instead if your question is specifically about Riemann's function.

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General formula for logarithmic cosine integral

I am trying to find a general expression for $$\int_0^{\pi/2} (\ln(\cos(x)))^n dx$$ for integer $n$, but have not been able to find it online or derive it. I have a good idea that the general form ...
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Asymptotically similar functions with opposite parity, were they considered, are they useful? Case of polynomials

So, can we transform an even function into an odd function and vice versa? Let's consider this method: Transformation even->odd: Suppose $f_{even}(x)$ is a function which satisfies the following ...
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Estimation of the Totient Summatory Function

The following statement appears on the Wikipedia page for the Totient Summatory Function: The Totient summatory function is defined by: $\Phi(n):=\sum_{k=1}^n\varphi(k),\ \ \ \ n\in\mathbb{N}$ Using ...
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A question on the relation between the complex zeroes of zeta function and the estimate of the error in PNT

I'm currently working on the following problem from Analytic Number Theory. Assume that $\psi(x)-x=\mathcal{O}(x^a)$, for some $1/2<a<1$, where $\psi$ is the Chebyshev function. I would like to ...
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The average value of $\zeta(1/2+it)$ at Gram points is 2

It is stated in $\textit{Riemann’s Zeta Function}$ (H.M. Edwards) in the footnote of p126 that the average value of $\zeta(1/2+it)$ at Gram points is 2 - he cites Section 11.1 of the paper ON VAN DER ...
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Drinfeld associators expansion to higher weights

Does anyone know where to get higher orders (for weight >10) expansion in Drinfeld associators. (the generating function for MZVs) The answer might be in the form of (4.5) in https://arxiv.org/pdf/...
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Counting numbers only containing primes in given set of primes

Suppose $\mathbb P$ is a set of primes, and more specifically I'm interested in infinite $\mathbb P$ but satisfying $\sum _{p\in \mathbb P}\frac {1}{p}<\infty .$ How can I evaluate \[ \sum _{n\...
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Ordinate of zeros of zeta Riemann function

In the book: Multiplicative Number Theory, Davenport, we see that there is a bound for ordinates of zeros of zeta Riemann function ($s=\beta+i\gamma$ is zero of zeta function) My question is that how ...
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Proof that $(1-\zeta(1-p,p))/p$ is an integer?

For all primes $p>2$, does the following identity $$\frac{1-\zeta(1-p,p)}{p}\in\mathbb{Z}$$ hold such that $\zeta(s,a)$ denotes the Hurwitz zeta function? If so, how would I go about formulating a ...
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Proving an identity for Riemann zeta type sum

I have asked here for a closed form of the sum $$\sum_{n=1}^∞\left(\frac{1}{n^2+\alpha^2}\right)^a.$$ However, it was suggested in the comments that even though a closed form might not be possible, ...
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