# Questions tagged [upper-bounds]

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### Supremum or upper bound of bivariate function involving logarithms and combinatorial coefficients or the gamma function over a region of the integers

This is a repost from MSE because I got no answers there.
I have been trying to find the supremum of this bivariate function over a specific region. However, the expressions that I get are horrible. I ...

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### Expectation of the inner product of a subset of two random orthonormal vectors

Setting: Consider sampling two orthonormal vectors $\mathbf{u},\mathbf{v}$ in $\mathbb{R}^p$ (where $p\ge2$) from a "uniform" distribution over the $p$-dimensional sphere (alternatively, ...

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### Convergence in expectation of a discontinuous function

Consider a random variable $X\in \mathbb{R}^d$. Let ${\theta_m}$ be a sequence of real numbers that converge to $\theta$. Let $f(x,y)$ be a function that is not continuous. To be specific, fix, $x=a$, ...

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### Is it possible to bound Mertens function $M(n)$ from an inclusion-exclusion formulation?

In this post I proposed a formulation of Mertens function $M(n)$ using the inclusion-exclusion principle, as follows:
$$M(n)=-\pi\left(n\right)+\left(\sum_{p_{i}<\sqrt{n}}\pi\left(\lfloor\frac{n}{...

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### Bound the probability that a point belongs to a set

Let $(a_k)_{k \geq 1}$ be random variables taking values on a finite subset $B$. Assume that
$$
(1) \quad \Pr\Big (\lim_{n\rightarrow +\infty}d(\frac{1}{n}\sum_{k=1}^n 1_{[a_k = b]}, [v_\ell(b,\...

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### Bound the expectation of an average

Let $(a_n)_{n \geq 1}$ be random variables taking values on a finite subset $B$. Assume that $\nu_l(b) \le P[a_n = b\mid a_1,\ldots,a_{n-1}] \le \nu_u(b)$ almost surely for every $n \ge 1$ and $b \in ...

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### Tools to prove lower bounds in analytic number theory

Perron's formula and related methods are used to relate statements such as the Riemann hypothesis to upper bounds of functions occurring in analytic number theory. For example, Perron's formula is ...