View source for Halász mean value theorem

Considerable interest has been devoted to the problem of obtaining conditions for arithmetic functions (cf. [[Arithmetic function|Arithmetic function]]), in particular for multiplicative functions (cf. [[Multiplicative arithmetic function|Multiplicative arithmetic function]]), that guarantee the existence of a mean value

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A strong motivation for this interest was the famous Erdös–Wintner conjecture (see [[#References|[a2]]]): Any multiplicative function assuming only the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100103.png" /> possesses a mean value.

Around 1961, theorems of H. Delange (see [[Delange theorem|Delange theorem]]) and E. Wirsing (see [[Wirsing theorems|Wirsing theorems]]) gave a satisfactory answer for multiplicative functions with non-zero mean value. However, a general mean value theorem, containing a proof of the prime number theorem (cf. also [[De la Vallée-Poussin theorem|de la Vallée-Poussin theorem]]) and of the Erdös–Wintner conjecture, was only given by Wirsing in 1967 (see [[#References|[a7]]]) and G. Halász in 1968 (see [[#References|[a3]]]).

A simple form of Halász' theorem reads as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100104.png" /> is a multiplicative function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100105.png" />, then there exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100107.png" />, and a slowly oscillating function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100108.png" />, satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100109.png" />, such that

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In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001011.png" /> is real-valued, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001013.png" />, and so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001014.png" /> exists. For the proof, Halász used the classical method of complex integration (cf. [[Contour integration, method of|Contour integration, method of]]) in a very skilful way.

A more precise formulation (see [[#References|[a3]]], Satz 2) is as follows. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001015.png" /> is multiplicative and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001016.png" />.

i) If the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001017.png" /> diverges for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001018.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001019.png" /> exists and is equal to zero.

ii) If for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001020.png" /> the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001021.png" /> is convergent, then, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001022.png" />,

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001026.png" />.

The unpleasant condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001027.png" /> in the theorem was (partly) removed, and remainder estimates were given, in, e.g., [[#References|[a4]]], [[#References|[a8]]], [[#References|[a9]]]. K.-H. Indlekofer [[#References|[a6]]] gave a version of the theorem for the class of  "uniformly summable"  multiplicative functions. A uniformly summable multiplicative function is a multiplicative function satisfying

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and

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In 1988, A. Mačiulis gave a version of Halász' theorem with remainder term. Elementary proofs of the Halász theorem were published in [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Daboussi,   K.-H. Indlekofer,   "Two elementary proofs of Halász's theorem"  ''Math. Z.'' , '''209'''  (1992)  pp. 43–52</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Erdös,   "Some unsolved problems"  ''Michigan Math. J.'' , '''4'''  (1957)  pp. 291–300</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Halász,   "Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen"  ''Acta Math. Acad. Sci. Hung.'' , '''19'''  (1968)  pp. 365–403</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Halász,   "On the distribution of additive and the mean values of multiplicative arithmetic functions"  ''Studia Sci. Math. Hung.'' , '''6'''  (1971)  pp. 211–233</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G. Halász,   "Remarks to my paper  "On the distribution of additive and the mean values of multiplicative arithmetic functions" "  ''Acta Math. Acad. Sci. Hung.'' , '''23'''  (1972)  pp. 425–432</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K.-H. Indlekofer,   "Remark on a theorem of G. Halász"  ''Archiv Math.'' , '''36'''  (1981)  pp. 145–151</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E. Wirsing,   "Das asymptotische Verhalten von Summen über multiplikative Funktionen, II"  ''Acta Math. Acad. Sci. Hung.'' , '''18'''  (1967)  pp. 414–467</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Parson,   J. Tull,   "Asymptotic behavior of multiplicative functions"  ''J. Number Th.'' , '''10'''  (1978)  pp. 395–420</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M.I. Tuljaganova,   "A generalization of a theorem of Halász"  ''Izv. Akad. Nauk UzSSR'' , '''4'''  (1978)  pp. 35–40; 95  (In Russian)</TD></TR></table>