On some solutions of a functional equation related to the partial sums of the Riemann zeta function

In this paper, we prove that infinite-dimensional vector spaces of α-dense curves are generated by means of the functional equations f(x)+f(2x)+⋯+f(nx)=0, with n≥2, which are related to the partial sums of the Riemann zeta function. These curves α-densify a large class of compact sets of the plane for arbitrary small α, extending the known result that this holds for the cases n=2,3. Finally, we prove the existence of a family of solutions of such functional equation which has the property of quadrature in the compact that densifies, that is, the product of the length of the curve by the nth power of the density approaches the Jordan content of the compact set which the curve densifies.

The Zeros of Riemann Zeta Partial Sums Yield Solutions to f(x) + f(2x) + · · · + f(nx) = 0

This paper proves that every zero of any n th , n ≥ 2, partial sum of the Riemann zeta function provides a vector space of basic solutions of the functional equation f(x)+f(2x)+⋯+f(nx)=0,x∈R . The continuity of the solutions depends on the sign of the real part of each zero.

A note on the real projection of the zeros of partial sums of Riemann zeta function

This paper proves that the real projection of each simple zero of any partial sum of the Riemann zeta function ζn(s):=∑nk=11ks,n>2 , is an accumulation point of the set {Res : ζ n (s) = 0}.

Computing the zeros of the partial sums of the Riemann zeta function

In this paper, we introduce a formula for the exact number of zeros of every partial sum of the Riemann zeta function inside infinitely many rectangles of the critical strips where they are situated.

Comparison Between Dcrit Considering the Abrupt Variation and Inflexion in the Concrete Mercury Intrusion Porosimetry Curve

Mercury intrusion porosimetry (MIP) has been widely used to evaluate the quality of concrete through the pore size distribution parameters. Two of these parameters are the critical pore diameter (Dcrit) and the percentage of the most interconnected net of pores compared to the total volume of pores. Some researchers consider Dcrit as the diameter obtained from the inflexion point of the cumulative mercury intrusion curve while others consider Dcrit as the diameter obtained from the point of abrupt variation in the same curve. This study aims to analyze two groups of concretes of varying w/c ratios, one cast with pozzolanic cement and another with high initial strength cement, in order to determine which of these diameters feature a better correlation with the quality parameters of the concretes. The concrete quality parameters used for the evaluations were (1) the w/c ratios and (2) chloride diffusion coefficients measured at approximately 90 days. MIP cumulative distributions of the same concretes were also measured at about 90 days, and Dcrit values were determined (1) from the point of abrupt variation and alternatively, (2) from the inflexion point of each of these plots. It was found that Dcrit values measured from the point of abrupt variation were useful indicators of the quality of the concrete, but the Dcrit values based on the inflexion points were not. Hence, it is recommended that Dcrit and the percentage of the most interconnected volume of pores should be obtained considering the point of abrupt variation of the cumulative curve of pore size distribution.