Related papers: Improving an inequality for the divisor function

In this short note, we improve the famous Reid Inequality related to linear operators.

We extend a Gaussian functional inequality to a countable product of Gaussian measures. This inequality improves on the classical Poincare inequality for Gaussian measures. As an application, we prove that First Passage Percolation has…

We establish asymptotic formulae for various correlations involving general divisor functions $d_k(n)$ and partial divisor functions $d_l(n,A)=\sum_{q|n:q\leq n^A}d_{l-1}(q)$, where $A\in[0,1]$ is a parameter and $k,l\in\mathbb{N}$ are…

We prove some distribution results for the $k$-fold divisor function in arithmetic progressions to moduli that exceed the square-root of length $X$ of the sum, with appropriate constrains and averaging on the moduli, saving a power of $X$…

In this paper, we consider the fractional sum of the divisor functions. We can improve previous results considered by Bordell\'{e}s \cite{Bo} and Liu-Wu-Yang \cite{LWY}.

Finite sample bounds on the estimation error of the mean by the empirical mean, uniform over a class of functions, can often be conveniently obtained in terms of Rademacher or Gaussian averages of the class. If a function of n variables has…

The upper bound and the lower bound of average numbers of divisors of Euler Phi function and Carmichael Lambda function are obtained by Luca and Pomerance (see \cite{LP}). We improve the lower bound and provide a heuristic argument which…

Pisier's inequality is central in the study of normed spaces and has important applications in geometry. We provide an elementary proof of this inequality, which avoids some non-constructive steps from previous proofs. Our goal is to make…

Using sieves and elementary manipulations, we show that the signs of partial sums of the Liouville function over divisors are in a strong sense equally distributed.

We exploit the properties of a sequence of functions that approximate the divisor functions and combine them with an analytical formula of a delta-like sequence to give a new proof of a theorem of Gronwall on the asymptotic of the divisor…

We extend the sum-of-divisors function to the complex plane via the Gaussian integers. Then we prove a modified form of Euler's classification of odd perfect numbers.

We introduce a modification of the linear sieve whose weights satisfy strong factorization properties, and consequently equidistribute primes up to size $x$ in arithmetic progressions to moduli up to $x^{10/17}$. This surpasses the level of…

This paper aims to characterize the function appearing in the weighted Hermite-Hadamard inequality. We provide improved inequalities for the weighted means as applications of the obtained results. Modifications of the weighted…

We discuss the Pointwise Ergodic Theorem for the Gaussian divisor function $d(n)$, that is, for a measure preserving $\mathbb Z[i]$ action $T$, the limit $$\lim_{N\rightarrow \infty} \frac{1}{D(N)} \sum _{\mathscr{N} (n) \leq N} d(n)…

In this paper we shall review the common problems associated with Piecewise Linear Separation incremental algorithms. This kind of neural models yield poor performances when dealing with some classification problems, due to the evolving…

We consider the upper bound of Piltz divisor problem over number fields. Piltz divisor problem is known as a generalization of the Dirichlet divisor problem. We deal with this problem over number fields and improve the error term of this…

For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence.

The error term in the approximate functional equation for exponential sums involving the divisor function will be improved under certain conditions for the parameters of the approximate functional equation.

In this paper we propose a method for proving some exponential inequalities based on power series expansion and analysis of derivations of the corresponding functions. Our approach provides a simple proof and generates a new class of…

In this paper we give a short, new proof of a natural generalization of Gerzon's bound. This bound improves the Delsarte, Goethals and Seidel's upper bound in a special case. Our proof is a simple application of the linear algebra bound…