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Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
1
vote
1
answer
173
views
Is this function concave or convex? [closed]
let $g_{n,\gamma}(\sigma)$ be the function defined as the following
$$
g_{n,\gamma}(\sigma)= \left(\frac{(\sigma-1)^2 +\gamma^2}{\sigma^2
+\gamma^2} \right)^{n/2} T_n\left( \frac{\sigma(\sigma-1)
+\g …
0
votes
0
answers
80
views
Comparison of two functions
Given a function $f$ from $R^2$ to $R$ satisfying tha following:
$1)$ $f$ is a convex function which vanishes on $(0,1)$ and on $(1,0).$
$2)$ $f$ is a decreasing function on $x$ and on $y$ and $f …
0
votes
1
answer
125
views
Discussion for the sign of a specific sum
Given a modular form $f$ of an even weight $k$ for the full modular group. Let $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ For a fixed positive integers $a$ and $b,$
I want to dis …
1
vote
Accepted
Discussion for the sign of a specific sum
Using The Euler product, we can express the sum
$$\sum_{\substack{l=1\\\gcd(l,a)=\gcd(l,b)=1}}^{+\infty}\left(\frac{\lambda_f(l)}{l^{3/4}}\right)^3$$ as an infinite product over prime numbers so that …