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Register a new userConjectures involving a generalization of the sums of powers of integers
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Authors
Andrei K. Svinin
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Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.
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In this note, we extend the definition of multiple harmonic sums and apply their stuffle relations to obtain explicit evaluations of the sums $R_n(p,t)=sum olimits_{m=0}^n m^p H_m^t$, where $H_m$ are harmonic numbers. When $tle 4$ these sums were first studied by Spiess around 1990 and, more recently, by Jin and Sun. Our key step first is to find an explicit formula of a special type of the extended multiple harmonic sums. This also enables us to provide a general structural result of the sums $R_n(p,t)$ for all $tge 0$.
We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.
We show that in any two-coloring of the positive integers there is a color for which the set of positive integers that can be represented as a sum of distinct elements with this color has upper logarithmic density at least $(2+sqrt{3})/4$ and this is best possible. This answers a forty-year-old question of ErdH{o}s.
We consider several families of binomial sum identities whose definition involves the absolute value function. In particular, we consider centered double sums of the form [S_{alpha,beta}(n) := sum_{k,;ell}binom{2n}{n+k}binom{2n}{n+ell} |k^alpha-ell^alpha|^beta,] obtaining new results in the cases $alpha = 1, 2$. We show that there is a close connection between these double sums in the case $alpha=1$ and the single centered binomial sums considered by Tuenter.
We develop novel techniques which allow us to prove a diverse range of results relating to subset sums and complete sequences of positive integers, including solutions to several longstanding open problems. These include: solutions to the three problems of Burr and ErdH{o}s on Ramsey complete sequences, for which ErdH{o}s later offered a combined total of $350; analogous results for the new notion of density complete sequences; the solution to a conjecture of Alon and ErdH{o}s on the minimum number of colors needed to color the positive integers less than $n$ so that $n$ cannot be written as a monochromatic sum; the exact determination of an extremal function introduced by ErdH{o}s and Graham on sets of integers avoiding a given subset sum; and, answering a question reiterated by several authors, a homogeneous strengthening of a seminal result of Szemeredi and Vu on long arithmetic progressions in subset sums.
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