scholarly journals On integrals associated with the free particle wave packet

2020 ◽  
Vol 26 (2) ◽  
pp. 257-262
Author(s):  
Alexander E. Patkowski

AbstractWe discuss some properties of integrals associated with the free particle wave packet, {\psi(x,t)}, which are solutions to the time-dependent Schrödinger equation for a free particle. Some noteworthy discussion is made in relation to integrals which have appeared in the literature. We also obtain formulas for half-integer arguments of the Riemann zeta function.

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .


1994 ◽  
Vol 37 (2) ◽  
pp. 278-286 ◽  
Author(s):  
C. Yalçin Yildirim

AbstractA relation between the zeros of the partial sums and the zeros of the corresponding tails of the Maclaurin series for ez is established. This allows an asymptotic estimation of a quantity which came up in the theory of the Riemann zeta-function. Some new properties of the tails of ez are also provided.


1932 ◽  
Vol 28 (3) ◽  
pp. 273-274 ◽  
Author(s):  
E. C. Titchmarsh

It was proved by Littlewood that, for every large positive T, ζ (s) has a zero β + iγ satisfyingwhere A is an absolute constant.


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