Related papers: Dual Smale's mean value conjecture for $n=7$

In this paper, we study the multiple $L$-values and the multiple zeta values of level $N$. We set up the algebraic framework for the double shuffle relations of the multiple zeta values of level $N$. Using the regularized double shuffle…

Prime numbers are fascinating by the way they appear in the set of natural numbers. Despite several results enlighting us about their repartition, the set of prime numbers is often informally qualified as misterious. In the present paper,…

An elementary recursive relation for M$\ddot{\mathrm{o}}$bius function $\mu (n)$ is introduced by two simple ways. With this recursive relation, $\mu (n)$ can be calculated without directly knowing the factorization of the $n$. $\mu (1)…

We present a new proof of generalized Flett's mean value theorem due to Pawlikowska (from Demonstratio Math. 1999) using only the original Flett's mean value theorem. Also, a Trahan-type condition is established in general case.

We give a reduction of Donovan's conjecture for abelian groups to a similar statement for quasisimple groups. Consequently we show that Donovan's conjecture holds for abelian $2$-groups.

We obtain an asymptotic formula for the mean value of the function $\tau_k(n)$, which is the number of solutions of the equation $x_1...\, x_k=n$ in natural numbers $x_1,..., x_k$ in some special sequences of natural numbers.

Let $(\epsilon_i)$ be a Rademacher sequence, i.e., a sequence of independent and identically distributed random variables satisfying $P(\epsilon_i=1)=P(\epsilon_i=-1)=1/2$. Set $S_n=a_1\epsilon_1+\cdots+a_n\epsilon_n$ for…

We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification.

We establish a short exact sequence about depth-graded motivic double zeta values of even weight relative to $\mu_2$. We find a basis for the depth-graded motivic double zeta values relative to $\mu_2$ of even weight and a basis for the…

In order to investigate minimal sufficient conditions for an abstract integral to belong to the convex hull of the integrand, we propose a system of axioms under which it happens. If the integrand is a continuous $R^n$-valued function over…

We extend a result by Ikeda and Suriajaya (2025) to find the asymptotic behaviour of the average number of representations of an integer $n$, over multiples of a fixed $q\ge 2$, as a sum of two prime $k$-th powers, for $k\ge 2$.

In this paper we propose a general methodology, based on multiple testing, for testing that the mean of a Gaussian vector in R^n belongs to a convex set. We show that the test achieves its nominal level, and characterize a class of vectors…

The Ramanujan Machine project predicts new continued fraction representations of numbers expressed by important mathematical constants. Generally, the value of a continued fraction is found by reducing it to a second order linear difference…

Given a multiplicative function f satisfying |f(n)| <= 1 for all n, the authors study the problem of obtaining explicit upper bounds on the mean-value 1/x |sum_{n <= x} f(n)|.

Given an integer g, g > 1, an integer w, -1 < w <g - 1, and a set of g distinct numbers, c_1, ..., c_g, we present a conjectured identity for Stirling numbers of the first kind. We have proven all the equalities in case g < 7; and for the…

We obtain new mean value theorems for exponential sums with very smooth numbers, which provide a power saving against the trivial bound in region where previous bounds do not apply.

In this paper, we state as a conjecture a vector-valued Hopf-Dunford-Schwartz lemma and give a partial answer to it. As an application of this powerful result, we prove some Fe fferman-Stein inequalities in the setting of Dunkl analysis…

In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)= \sum_{j=1}^\infty a_j n_j^{-s}$ with positive coefficients for which the counting function $A(x) = \sum_{n_{j}\le x}a_{j}$ satisfies $A(x)=\rho x +…

We revisit the Kahn-Kalai conjecture, recently proved in striking fashion by Park and Pham, and present a slightly reformulated simple proof which has a few advantages: (1) it works for non-uniform product measures, (2) it gives…

For the mean vector test in high dimension, Ayyala et al.(2017,153:136-155) proposed new test statistics when the observational vectors are M dependent. Under certain conditions, the test statistics for one-same and two-sample cases were…