RECENT RESULTS ON STIELTJES CONSTANTS ADDENDUM TO HIGHER DERIVATIVES OF THE HURWITZ ZETA FUNCTION. 1. Introduction

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1 RECENT RESULTS ON STIELTJES CONSTANTS ADDENDUM TO HIGHER DERIVATIVES OF THE HURWITZ ZETA FUNCTION DOMINIC LANPHIER. Introduction Much recent significant wor on the arithmetic and analytic properties of Stieltjes constants has been done in a series of papers such as [], [2], and [3]. Here we briefly review some of these results and relate them to results of this thesis. Mar Coffey is thaned for bringing our attention to this wor. The omission of these citations is regretted. The Stieltjes constants have been well-studied from a variety of viewpoints. In particular, their analytic and asymptotic properties have been investigated in [] and various arithmetic properties investigated in [2], and [3]. As presented in [2], the Stieltjes constants can be defined as γ a = lim n n j=0 ln j + a j + a ln+ n + a, + and they occur as coefficients in the Laurent expansion about s = of the Hurwitz zeta function ζs, a = s + n γ n a s n, s =. n!

2 2 RECENT RESULTS ON STIELTJES CONSTANTS One of the consequences of the methods of the thesis is that by looing at derivatives of ζs, a, certain relations concerning the Stieltjes constants can be found. These relations can essentially be found in [2]. There, several very general results are given which have numerous relations of Stieltjes constants as straightforward consequences. For example, in 3.27 of [2], a relation of the Stieltjes constants to series involving logarithms is found. In particular, it is shown that γ a γ b = ln n + a n + a ln n + b. n + b Further, various general summation relations for Stieltjes constants are shown, such as the following. Proposition Proposition, [2]. If Rea > 0 and Reb > 0 then Γb n! γ n+a γ n+ b = ln. Γa In the next section of [2], various relations to derivatives of certain Dirichlet L-series are obtained. A very general result encompassing many relations of the numbers γ a is shown. More precisely, the following result for odd Dirichlet characters is given.

3 RECENT RESULTS ON STIELTJES CONSTANTS 3 Proposition 2 Proposition 3, [2]. Suppose that χ is a nonprincipal Dirichlet character of conductor so that χ =. Then χ mγ = L ln lnu + γ = χ 0 e u me mu du ln = π /2 ln2π + γ mχ m π /2 ln χ mψ χ mψ Γ χ m m ln χ mψ. Here L s is the Dirichlet L-function Ls, χ = n= χ nn s. The analogous result for even Dirichlet characters is also given. Proposition 3 Proposition 4, [2]. Suppose that χ is a nonprincipal Dirichlet character of conductor so that χ =. Then χ mγ = L + ln lnu + γ = χ 0 e u me mu du ln = /2 2ln2π + γ ln ln χ mψ. Γ χ m m χ mψ χ mψ χ mζ 0, m These are every general results, and they encompass the Stieltjes constant identities found in the thesis. In particular, several explicit identities involving the first Stieltjes constant can be shown from this formula. For example, taing = 3 see 3.20 of [2]

4 4 RECENT RESULTS ON STIELTJES CONSTANTS we get 2 γ γ 3 3 = 3 π Γ ln2π + γ 3 ln 3 Γ ln3 and 3.2 of [2] is 3 Γ 4 γ γ = π ln8π + γ 2 ln 4 4 Γ. 3 4 Taing γ m = γ m, from 3.28 of [2] and also Proposition 3 of [] and Proposition 5. of [3] and Corollary 3 of [2] for the = case we have q r γ q r= = γ + q ln+ q + + q r r r=0 ln r qγ r. Further, expressions for γ a for 2 are also obtained, such as.2 and.22 in [2]. These expressions give γ 2 a and, more generally, γ n a in terms of certain values of derivatives of Hurwitz zeta functions. For example, Proposition 8 in [2] is, for Rea > 0, γ 2 a = 3 ln3 a + + ln2 a a = ζ +, a + + 2! s +, 2ζ +, a s +, 3ζ +, a +! where sn, m are the Stirling numbers of the first ind.

5 RECENT RESULTS ON STIELTJES CONSTANTS 5 There also is a general expression for such Stieltjes constants given in Proposition 6. of [3] as follows, γ m = m! m l= ln2 m + B m l+ ln m l 2 m l +! l! n= 2 n+ n= 2 n+ n n ln l + + = n n ln m+ + + = where the B m are the Bernoulli numbers. B m+ m + lnm+ 2 In this thesis, properties of the Hurwitz zeta function were investigated and higher derivatives of the Hurwitz zeta function at s = 0 were evaluated. Several relations among the Stieltjes constants were then shown and these relations can be encompassed in the general results of [2]. References [] M.W. Coffey, New results on the Stieltjes constants: Asymptotic and exact evaluation, J. Math. Anal. Appl , [2] M.W. Coffey, On representations and differences of Stieltjes coefficients, and other relations, Rocy Mountain J. Math. 4 no. 6 20, [3] M.W. Coffey, New summation relations for Stieltjes constants, Proc. Royal Soc. A ,