Tagged Questions
3
votes
3answers
92 views
The number $ \frac{(m)^{(k)}(m)_k}{(1/2)^{(k)} k!}$
For a real number $a$ and a positive integer $k$, denote by $(a)^{(k)}$ the number $a(a+1)\cdots (a+k-1)$ and $(a)_k$ the number
$a(a-1)\cdots (a-k+1)$. Let $m$ be a positive integer $\ge k$. Can ...
0
votes
1answer
20 views
Ordinary generating function for Bernoulli polynomial
I know the exponential generating function for the Bernoulli polynomial $B_n(x)$:$$\frac{te^{tx}}{e^t-1}=\sum_{n=0}^\infty B_n(x)\frac{t^n}{n!}.$$
But is there an ordinary generating function? i.e a ...
1
vote
1answer
52 views
Coefficients of powers of the theta function
Let $q=\exp(2 \pi i z)$ and $$\theta(z)=\sum_{n=-\infty}^\infty q^{n^2}.$$
Now, I shall show that the powers of $\theta$ are given by
$$\theta(z)^r = \sum_{n=0}^\infty S_r(n) q^n$$
where $S_r(n)$ ...
4
votes
1answer
119 views
Dirichlet L-series and Gamma function question
Could someone help me, please, with this exercise?
Consider a sequence of complex numbers $\{a_n\}$ such that $a_n=a_m $ iff $ n\cong m $ mod $q$ for some positive integer $q$.
Define the ...
1
vote
1answer
73 views
Dilogarithm Identities
Is there a cleaner way to write:
$$
f(x) = \operatorname{Li}_2(i x) - \operatorname{Li}_2(-i x)
$$
in terms of simpler functions? I don't know enough about dilogarithms, and the basic identities I see ...
2
votes
1answer
102 views
Are these numbers $h_{r,s}$ irrational?
I came across these numbers in my work some time ago. This type of expressions do not exist in closed form (not to confuse with Vandermonde convolution), I already know that. To simplify I denote
...
0
votes
0answers
62 views
When inequality for binomial coefficients is true?
I've asked similar question here Inequality for binomial coefficients, but with slightly different assumptions. I am curious what happend if $m, k$ are fixed.
Let $m \leq n, n \leq N$ and $0\leq k ...
5
votes
0answers
116 views
Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p} $ been discussed anywhere?
In an older fiddling with the gamma-function (expanding on the idea of sums of consecutive like-powers of logarithms, similarly as the bernoulli-polynomials for the sums of like powers of consecutive ...
4
votes
1answer
109 views
Nagura's paper--can we substitute for the original upper bound?
This question concerns two results about primes. The first is J. Nagura's 1952 result, that there is a prime on the interval $[x, (1+1/5)x] $ for $x> 2103,$ which depends on the result derived ...
3
votes
1answer
91 views
Minimal $x$ for which $\phi(k) > n$ for all $k > x$
It's well-known that
$$
\liminf_n\frac{\varphi(n)\log\log n}{n}=e^{-\gamma}
$$
and there exists an effective version
$$
\varphi(n)>\frac {n}{e^\gamma\log\log n+\frac{3}{\log\log n}}
$$
valid for ...
4
votes
1answer
179 views
How to prove Gauss's Digamma Theorem?
Here $\psi(z)$ is digamma function, $\Gamma(z)$ is gamma function. $$\psi(z)=\frac{{\Gamma}'(z)}{\Gamma(z)},$$
For positive integers $m$ and $k$ (with $m < k$), the digamma function may be ...
4
votes
2answers
139 views
Could you give a application of a special function on number theory or analysis?
With the best effort i have ever taken, i couldn't find a application of a special function on number theory or analysis on the internet. By the way, why is the applications of special functions in ...
1
vote
1answer
195 views
Verifying identities for Riemann zeta function
I ran across these two problems, while reading a text on number theory. The problem states: "Verify the following identities". What does "verify" mean in this context, and what strategies can I employ ...
19
votes
1answer
452 views
Upper bound on differences of consecutive zeta zeros
The average gap $\delta_n=|\gamma_{n+1}-\gamma_n|$ between consecutive zeros $(\beta_n+\gamma_n i,\beta_{n+1}+\gamma_{n+1}i)$ of Riemann's zeta function is $\frac{2\pi}{\log\gamma_n}.$ There are many ...
2
votes
1answer
122 views
Reason behind the reciprocity of series
This question may appear to be a silly one for experts.
From long back I have been observing all kinds of series but every-series contain a reciprocal part, I mean the " one over something " , is ...
7
votes
2answers
291 views
Why is this sum equal to the Logarithmic Integral?
I am using this sum:
$$\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}\left((-1)^{k-1} (n-1) + \sum_{j=1}^{k-1}\frac{(-1)^{j+k-1}n (\log n)^j}{j!}\right)$$
Empirically, this is precisely equal to
...
5
votes
2answers
177 views
Lerch-$\small \zeta(\varphi,0,-n)$ of integer *n* purely real and imaginary ($\small \zeta_\varphi (-n)^2 $ is real) for $\small n \ge 2$?
Are the Lerch-$\zeta(\varphi,0,-n) $ of integer n (for shortness I use the notation of my earlier question $\small \zeta_\varphi(-n)$) periodically purely real and imaginary: $\zeta_\varphi (-n)^2 $ ...
2
votes
4answers
321 views
Generalization $\zeta_\varphi(s)=\sum_{k=0}^\infty {\exp(I\varphi*k) \over (1+k)^s} $
This is more a reference-request for some fiddling/exploration with the $\zeta$-function. In expressing the $\zeta$ and the alternating $\zeta$ (="$\eta$") in terms of matrixoperations I asked myself, ...
12
votes
3answers
1k views
Analytic continuation- Easy explanation?
Today, as I was flipping through my copy of Higher Algebra by Barnard and Child, I came across a theorem which said,
The series $$ 1+\frac{1}{2^p} +\frac{1}{3^p}+...$$ diverges for $p\leq 1$ and ...
20
votes
1answer
427 views
Feeding real or even complex numbers to the integer partition function $p(n)$?
Like most people, when I first encountered $n!$ in grade school, I graphed it, then connected the dots with a smooth curve and reasoned that there must be some meaning to $\left(\frac43\right)!$ — ...
5
votes
1answer
322 views
Where are this kind of series used, $\vartheta_{4}(0,e^{\alpha \cdot z})$?
In my recent explorations I stumbled upon the following series
$$
\vartheta_{4}(0,e^{\alpha \cdot z})=1+2\sum_{k=1}^{\infty} (-1)^{k}\cdot e^{\alpha \cdot z\cdot k^{2}} ; \alpha \in \mathbb{R}, z ...
3
votes
1answer
346 views
Efficiently calculating the logarithmic integral with complex argument
My number theory library of choice doesn't implement the logarithmic integral for complex values. I thought that I might take a crack at coding it, but I thought I'd ask here first for algorithmic ...
7
votes
4answers
473 views
Are there addition formulas for the Riemann Zeta function?
In particular for two real numbers $a$ and $b$, I'd like to know if there are formulas for $\zeta (a+b)$ and $\zeta (a-b)$ as a function of $\zeta (a)$ and $\zeta (b)$.
The closest I could find ...
1
vote
2answers
326 views
Legendre functions in number theory
I have heard that Legendre functions are important in number theory. Can any one tell me how?
The Legendre function of the first kind $P_s$ is defined by
\begin{eqnarray*}P_s(x) =& ...
