Related papers: A remark on Fine's arithmetic functions
We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite-dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable…
We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.
Inequalities play an important role in pure and applied mathematics. In particular, Jensen's inequality, one of the most famous inequalities, plays a main role in the study of the existence and uniqueness of initial and boundary value…
We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations…
This is the second of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression…
In this paper we prove the mean values of some multiplicative functions connected with the divisor function on the short interval of summation.
In this article, we generalize several fundamental results for arithmetic divisors, such as the continuity of the volume function, the generalized Hodge index theorem, Fujita's approximation theorem for arithmetic divisors and Zariski…
We study the shifted convolution sum of the divisor function and some other arithmetic functions.
We give a formula that relates the difference of the j-invariants with the Borcherds Phi-function, an automorphic form on the period domain for Enriques surfaces characterizing the discriminant divisor.
In this paper, we present a generalization of one of the theorems in [G. E. Andrews, Partitions with parts separated by parity, \textit{Annals of Combinatorics} \textbf{23}(2019), 241 - 248], and give its bijective proof. Further variations…
In this paper, a new identity for differentiable functions is derived. A consequence of the identity is that the author establishes some new general inequalities containing all of the Hermite-Hadamard and Simpson-like type for functions…
In this short note we present several infinite dimensional theorems which generalize corresponding facts from the finite dimensional differential inclusions theory.
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}.
In this paper, we establish some weighted fractional inequalities for differentiable mappings whose derivatives in absolute value are convex. These results are connected with the celebrated Hermite-Hadamard-Fejer type integral inequality.…
In this paper we look at normed spaces of differentiable functions on compact plane sets, including the spaces of infinitely differentiable functions originally considered by Dales and Davie. For many compact plane sets the classical…
Let $d,n$ be positive integers and $S$ be an arbitrary set of positive integers. We say that $d$ is an $S$-divisor of $n$ if $d|n$ and gcd $(d,n/d)\in S$. Consider the $S$-convolution of arithmetical functions given by (1.1), where the sum…
A characterization of multiplicative (and additive) arithmetical functions is given. Using this characterization, we show that the group of multiplicative arithmetical functions is isomorphic to the group of additive arithmetical functions.
The arithmetic function of two variables is defined. Some properties of the function are given along with the formula that is an analog of the so-called Mobius' inversion formula. A heuristic statement is suggested.
An extension of the method and results of A. Schwarz for evaluating the partition function of a quadratic functional is presented. This enables the partition functions to be evaluated for a wide class of quadratic functionals of interest in…
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc.…