17

I think that there is indeed some possibility to lower the bound, and this is something I've looked at seriously a few times. I spent a semester (in 2019) with the Computational Number Theory Group here at BYU trying to do it, without success. Let me outline some of the difficulties we found.
Issue 1. The main technique used in establishing the current ...

12

1. Normality of $\lambda^*$ in base $2$ is equivalent to the Chowla's conjecture. It is widely believed to be true, but very far from being proved.
2. Ergodicity of the Liouville function for Cesàro averages on $([N])_{N\in\mathbb{N}}$ (cf. Definition 2.5 in arXiv:1611.09338) is also equivalent to the Chowla's conjecture. Thanks to Will Sawin for clarifying ...

8

Let $s\in\mathbb{C}$ be any point with $\Re s>0$. The Dirichlet series of $\eta(s)$ converges, hence $L(s)$ converges if and only if $L^*(s)$ converges.
For $\sigma_0>1/2$, it is also known that $L(s)$ converges in the half-plane $\Re s>\sigma_0$ if and only if $\zeta(s)$ has no zero in that half-plane.
Combining the previous two statements, it ...

8

We show the following.
Theorem. For any $n \geq 6$, there is a permutation $\sigma \in S_n$ such that $\prod_{k=1}^n k^{\sigma(k)}$ is a square (respectively, a cube).
Proof. Let us first handle the case of squares. It is equivalent to find a subset $A$ of $\{1,\ldots,n\}$ with cardinality $r = \lceil \frac{n}{2} \rceil$ such that the product of the elements ...

3

The multiplicative group of $\mathbb{Z}_{pq} \cong \mathbb{Z}_{p} \times \mathbb{Z}_{q}$ is not cyclic, but is generated by $x =(a,1)$ and $y = (1,b)$ where $a$ is a primitive root mod $p$ and $b$ is primitive mod $q$. Whiteman's Lemma 1 shows that $xy$ has order the least common multiple of $p-1$ and $q-1$. Clearly $\langle x, xy\rangle$ is the whole ...

2

Take some large $n$ and consider the summands in $W(\sigma,t)$ for which $t\log p_k$ lie between $2n\pi$ and $2n\pi+\pi/3$. These are the primes $p_k$ which lie in the interval $[e^{2n\pi/t},e^{2n\pi/t+\pi/3t}]$. Writing $N=e^{2n\pi/t}$, prime number theorem implies that this interval contains $\gg\frac{N}{\log N}$ primes. For each such prime we have $\cos(t\...

2

I have a solution when $-1$ is a power of $q$ mod $n$ (which generalizes your observations) and when $n$ is prime and $q$ is 2.
We'll show that if there is a $z$ such that $z^q\equiv -1 \mod n$ where $n=\prod_i p_i^{q_i}$ then:
$$S_q(n)= \frac{1}{2} ( n - \prod_i p_i^{q_i-\lceil{ q_i/q }\rceil}) $$
In particular, this is a generalization of both your ...

1

A few references:
Rudolf Lidl, Harald Niederreiter
Finite Fields
Cambridge University Press, 1997
has a chapter on linear recurring sequences.
Introduction to Finite Fields and Their Applications
by the same authors is more approachable, the first being a research monograph.
There is also
Cryptographic Boolean Functions and Applications by
Thomas Cusick and ...

1

The slight issue with Will Sawin's answer above for the second part where he suggests Bertrand's postulate is the case where the prime in between n/2 and n equals n-6, n-4 or n-2. Therefore it seems better to use the fact that the largest prime gap less than 15x10^18 is 1510 so for odd 4x10^18 < n < 15x10^18 there is definitely a prime, p, such that n -...

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