Related papers: Explicit formulae for primes in arithmetic progres…

We parameterize solutions to the equality $\Phi_3(x)=\Phi_3(a_1)\Phi_3(a_2)\cdots\Phi_3(a_n)$ when each $\Phi_3(a_i)$ is prime. Our focus is on the special cases when $n=2,3,4$, as this analysis simplifies and extends bounds on the total…

Denote the sum of element orders in a finite group $G$ by $\psi(G)$ and let $C_n$ denote the cyclic group of order $n$. Suppose that $G$ is a non-cyclic finite group of order $n$ and $q$ is the least prime divisor of $n$. We proved that…

In this paper we give an asymptotic expansion including error terms for the number of cycles in homology classes for connected graphs. Mainly, we obtain formulae about the coefficients of error terms which depend on the homology classes and…

In this paper, we give explicit error bounds for the asymptotic expansion of the shifted distinct partition function $q(n +s)$ for any nonnegative integer $s$. Then based on this refined asymptotic formula, we give the exact thresholds of…

Fix coprime natural numbers $a,q$. Assuming the Prime $k$-tuple Conjecture, we show that there exist arbitrarily long arithmetic progressions of Carmichael numbers, each of which lies in the reduced residue class $a$ mod $q$ and is a…

We establish asymptotic formulas for sums of reciprocals of primes in arithmetic progressions, generalizing recent results on multiple Mertens evaluations by Tenenbaum, Qi, and Hu. Specifically, for any fixed constant $K>0$, we derive…

Let $p$ be a large odd prime, let $x=\log p)(\log\log p)^{3+\varepsilon}$ and let $q\ll\log\log p$ be an integer, where $\varepsilon>0$ is a small number. This note proves the existence of small prime quadratic residues and small prime…

Let $\gamma<1<c$ and $19(c-1)+171(1-\gamma)<9$. In this paper, we establish an asymptotic formula for exponential sums over Piatetski-Shapiro primes $p=[n^{1/\gamma}]$ in arithmetic progressions.

We use the main theorem of Boxer-Calegari-Gee-Pilloni (arXiv:1812.09269) to give explicit examples of modular abelian surfaces $A$ over $\mathbf{Q}$ without extra endomorhpisms such that $A$ has good reduction outside the primes 2, 3, 5,…

We generalize current known distribution results on Shanks--R\'enyi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function…

In this paper we consider error sums of the form \[\sum_{m=0}^{\infty} \varepsilon_m\Big( \,b_m\alpha - \frac{a_m}{c_m}\,\Big) \,,\] where $\alpha$ is a real number, $a_m$, $b_m$, $c_m$ are integers, and $\varepsilon_m=1$ or $\varepsilon_m…

We give an exact formula for the number of normal subgroups of each finite index in the Baumslag-Solitar group BS(p,q) when p and q are coprime. Unlike the formula for all finite index subgroups, this one distinguishes different…

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a,q)=1$. Denote by $\mathcal{P}_2(a,q)$ the least almost-prime…

We show that if A is a subset of {1,...,N} contains no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{1-o(1)} N). The approach is somewhat different from that used in arXiv:1007.5444.

Fix an integer $r\geq 3$. Let $q$ be a large positive integer and $a_1,...,a_r$ be distinct residue classes modulo $q$ that are relatively prime to $q$. In this paper, we establish an asymptotic formula for the logarithmic density…

Let $f : \mathbf{N} \rightarrow \mathbf{C}$ be a bounded multiplicative function. Let $a$ be a fixed integer (say $a = 1$). Then $f$ is well-distributed on the progression $n \equiv a \pmod{q} \subset \{1,\dots, X\}$, for almost all primes…

For an even Dirichlet character psi, we obtain a formula for L(1,psi) in terms of a sum of Dirichlet L-series evaluated at s=2 and s=3 and a rapidly convergent numerical series involving the central binomial coefficients. We then derive a…

In this paper, we prove the existence of an efficient algorithm for the computation of $q$-expansions of modular forms of weight $k$ and level $\Gamma$, where $\Gamma \subseteq SL_{2}({\mathbb{Z}})$ is an arbitrary congruence subgroup. We…

In this paper, we prove some computational results about equivariant cohomology over the cyclic group $C_{p^n}$ of prime power order. We show that there is an inductive formula when the dimension of the $C_p$-fixed points of the grading is…

We show that if A is a subset of {1,...,N} containing no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{3/4-o(1)} N).