Abstract. Zeros on and off the critical line are found for Titchmarsh's function As).Let s = a + it. E. C. Titchmarsh [1, pp. 240-244] With the help of programs for computing L and L' (Spira [3]), an exploratory computation of f(s) in the critical strip for 0 < t < 200 revealed the following zeros off the critical line:
Abstract. In this paper, we derive a simple error estimate for the Stirling formula and also give numerical coefficients.Stirling's formula is: log r(s) = (s -i) log 5 -s + i log 2zr(1)Formulas (1) and (2) This is the form given in the NBS Handbook, and is clearly poor near the imaginary axis. It follows, however, from this form, that if |arg s\ g ir/4, then the error in taking the first zzz terms of the asymptotic series is less in absolute value than the absolute value of the (zzz + l)st term. Another form of the remainder, valid for |arg s| î t -8, is derived in Whittaker and Watson [5, §13.6], but this remainder involves the Hurwitz zeta function, and has never been used for numerical estimates. An estimate for Rm, as given by (2), may be found in Nielsen [6, p. 208], and, expressed in current notation, is U?2m+2 \RJß)\ < (2zzz + l)(2zzz + 2) |5|2m+1 (cos (| arg s))2
AU complex zeros of each Hurwitz zeta function are shown to lie in a vertical strip. Trivial real zeros analogous to those for the Riemann zeta function are found. Zeros of two particular Hurwitz zeta functions are calculated.
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