SELBERG S CENTRAL LIMIT THEOREM ON THE CRITICAL LINE AND THE LERCH ZETA-FUNCTION. II

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1 SELBERG S CENRAL LIMI HEOREM ON HE CRIICAL LINE AND HE LERCH ZEA-FUNCION. II ANDRIUS GRIGUIS Deparmen of Mahemaics Informaics Vilnius Universiy, Naugarduko Vilnius, Lihuania rius.griguis@mif.vu.l Absrac: In his paper, we coninue he research on i heorems for he Lerch zeafuncion Lλ, α, /+i when he pair of parameers λ, α are near o 0, 0, 0, /, 0,, /, 0, 0. Key wolds: Cenral i heorem on he criical line, Lerch zea-funcion. Denoe by. Inroducion Φx = x e / d he sard normal disribuion funcion. As usual, le s = σ + i be a complex variable. For 0 λ 0 < α, he Lerch zea-funcion is given by e iλn Lλ, α, s = σ >, n + α s n=0 by analyic coninuaion elsewhere. For more informaion see [4]. In [3] we proved he following saemen. Le ε be a posiive fixed number, as small as we wan. Le i ε < λ < e / log or λ = or λ / < ε ε < α or α / < ε ; ii e / log λ < If we assume i, hen meas [0, ] : If we assume ii, hen meas [0, ] : ε < α or α / < ε. log Lλ, α, / + i log Lλ, α, / + i Eλ, α, = Φx,

2 A. GRIGUIS where Eλ, α, = i exp i + 3πi iλα + iα 4 λ. / i Similar i heorems were obained for arg Lλ, α, / + i. In his paper we prove i heorems for he Lerch zea-funcion Lλ, α, / + i when he pair of parameers λ, α are near o 0, 0, 0, /, 0,, /, 0, 0. For a clariy, we se up a able of 9 differen pairs of λ, α: 0,0 /,0,0 0,/ /,/,/ 0, /,, In [3] we invesigaed he i laws when λ, α are near o,,, /, /, /, /. In his paper, we deal wih 5 remaining cases from he able above. When λ, α are exacly,,, /, /, or /, / we have 3 L,, s = ζs, L, /, s = s ζs, L/,, s = s ζs, L/, /, s = s Ls, χ, where ζs denoes he Riemann zea-funcion, Ls, χ he Dirichles L-funcion χ is a Dirichle characer mod 4 wih χ3 =. Only in hese four cases excluding periodiciy by λ he Lerch zea-funcion has an Euler produc. In he fories of he las cenury A. Selberg proved he following i heorem meas [0, ] : log ζ/ + i = Φx he proof of he i heorem for he Riemann zea-funcion relies on he Euler produc i is no clear wheher i heorems can be obained for all Lerch zea-funcions, since in general he Lerch zea-funcion has no Euler produc. Firs we consider i heorems for log Lλ, α, / + i. We use noaion ν... = meas [0, ] :..., where in place of dos we mean some condiion saisfied by. By ε we always denoe a posiive fixed number, which can be as small as we wan. If he parameer λ is very close o 0 or α is very close o 0, hen he Lerch zea-funcion becomes very large. From he approximaion by a finie sum 5 see he nex secion we have ha Lλ, α, / + i = λ O /4 / λ α uniformly for 0 < λ, α, as, where λ denoes he fracional par of he number λ. Accordingly we can remove hese large erms obain cenral i heorems when λ is very close o 0 or α is very close o 0. Le Λλ, α, = i exp i + πi 4 iλα λ / i.

3 Eλ, α, = Consider he cases: a 0 < λ b c d e f 0 < λ 0 < λ SELBERG S CENRAL LIMI HEOREM 3 i exp i + 3πi iλα + iα 4 λ. / i log +ε log +ε α / log +ε 0 < α α, log +ε log +ε, log +ε, λ / 0 < α log +ε log λ < +ε e / log 0 < α e / log λ < 0 < α heorem. If we assume a or b hen 4 If we assume c hen If we assume d or e hen log Lλ, α, / + i Λλ, α, log +ε, log +ε. log Lλ, α, / + i Λλ, α, /α /+i If we assume f hen log Lλ, α, / + i /α /+i log Lλ, α, / + i Eλ, α /α /+i log +ε, For cases from a o f, similar heorems can be formulaed for arg Lλ, α, / + i. We formulae only one example below. heorem. If we assume a, hen arglλ, α, / + i Λλ, α, As an ineresing fac we noe ha L0, α, / + i = L, α, / + i, bu i laws differs when λ ends o 0, compare equaliies 4. he i laws remains rue if in he condiions i ii we replace ε by / log +ε. heorems are proved in he nex secion.

4 4 A. GRIGUIS. Proofs Firs we formulae several lemmas laer derive heorems. Lemma 3. For λ, α equal o 0,, 0, /, we have ha ν log Lλ, α, / + i Proof. Since L0, α, s = L, α, s, he proof follows from Lemma. in [3]. Similarly as in [], Lemma 3 can be exended o arg Lλ, α, /+i. he following lemma will be needed also for heorem. Lemma 4. If a sequence of disribuion funcions F n x converges weakly o coninuous disribuion funcion F x, hen his convergence is uniform in x, <. Proof. he proof of lemma can be found in Perov [5]. We consider how close are wo Lerch zea-funcions if heir parameers are also close. Recall ha i Eλ, α, = exp i + 3πi iλα + iα 4 λ / i Λλ, α, = i exp i + πi 4 iλα λ. / i Lemma 5. Le 0 < λ, λ, α, α. Le max λ λ, α α 3/4. hen Lλ, α, / + i Eλ, α, Λλ, α, α / i Lλ, α, / + i + Eλ, α, + Λλ, α, + α / i λ λ + α α + /4. Proof. he Lerch zea-funcion can be approximaed by a finie sum. We have see Garunkšis [] ha 5 Lλ, α, / + i = + 0 n / e iλn n + α /+i i exp i + πi 4 iλα + O /4 0 n / e iαn e π+πi/+iα n + λ / i λ / i

5 uniformly in λ α, 0 < λ, α. hus 6 n SELBERG S CENRAL LIMI HEOREM 5 Lλ, α, / + i Eλ, α, Λλ, α, α / i Lλ, α, / + i + Eλ, α, + Λλ, α, + α / i e iλn n + α /+i e iλn n + α /+i n n + α n + α /+i + e iα n+α λ n + λ / i e iα n+α λ n + λ / i n + λ n + λ / i n + O /4 := A + B + O /4. We consider he firs sum in formula 6. A = n + α / i e iλ n e inλ λ + α /+i α n n + α n + α / expinλ λ exp + i log + α α. n + α By aylor expansion of funcions e x log x we obain A n n + α / n λ λ + α α + λ λ α α. n + α he bounds n / 3/4, n 3/ < n / /4 n n n leads o A 3/4 λ λ + α α + 5/4 λ λ α α. Similarly, we derive ha he second sum in formula 6 is B 3/4 α α + λ λ + /4 α λ α λ + 5/4 λ λ α α + α λ α λ λ λ. he lemma is proved. Proof of heorem We proof only he case a, where λ, α, depending on, is close o 0,. Remaining cases are analogous. Recall ha L0,, / + i = ζ/ + i. If ζ/ + i 0, hen log Lλ, α, / + i Λλ, α, = log ζ/ + i + log + Lλ, α, / + i Λλ ζ/ + i ζ/ + i.

6 6 A. GRIGUIS From he las equaliy we see ha log Lλ, α, /+i Λλ, α, is near o log ζ/+i if ζ/ + i is no very small. We expec ha here are no many for which ζ/ + i is very small. For his reason we choose some monoone funcion K, which saisfies he following condiions: K + as + K. Accordingly, we divide he inerval [0, ] in o wo inervals: [0, /K [/K, ]. he second inerval we divide in o wo subses log ζ/ + i J = [/K, ] : < K I = [/K, ] : J. By Lemma 4 recall ha Φx denoes he sard normal disribuion funcion we see ha measj log ζ/ + i =ν < K + o as. For I, we have =Φ K + o = o, log Lλ, α, / + i Λλ, α, Lλ, α, / + i Λλ ζ/ + i = log ζ/ + i + log + O ζ/ + i Since I, by Lemma 5 we see ha Lλ, α, / + i Λλ ζ/ + i ζ/ + i exp K Lλ, α, / + i Λλ ζ/ + i exp K exp π /4 K λ / + λ + α + K log exp / /4 + log ε + o. And finally, for I, we have log Lλ, α, / + i Λλ, α, = log ζ/ + i + o.

7 SELBERG S CENRAL LIMI HEOREM 7 Now we can finish he proof. As, we obain ha log Lλ, α, / + i Λλ, α, ν = meas [0, /K : log Lλ, α, / + i Λλ, α, x + meas I : log Lλ, α, / + i Λλ, α, x + meas J : = meas I : log Lλ, α, / + i Λλ, α, x + o log ζ/ + i + o x + O meas J + o = ν log ζ/ + i x + o + o = Φ x + o. Proof of heorem is similar o he proof of heorem. Noe ha Lemma 3 proof of heorem can be rewrien for arg Lλ, α, / + i see he noe afer he proof of Lemma 3. References [] E. Bombieri D.A. Hejhal, On he disribuion of zeros of linear combinaions of Euler producs, Duke Mah. J , [] R. Garunkšis, Approximaion of he Lerch zea-funcion, Lih. Mah. J., , [3] R. Garunkšis, A. Griguis, A. Laurinčikas, Selberg s Cenral Limi heorem on he Criical Line he Lerch Zea-Funcion, New Direcions in Value Disribuion heory of zea L-Funcions, proceedings of Wurzburg Conference, Ocober 6-0, 008, Shaker Verlag, 009, [4] A. Laurinčikas R. Garunkšis, he Lerch Zea-Funcion, Kluwer Academic Publishers 00. [5] V.V. Perov, Limi heorems for Sums of Independen Rom Variables, Nauka, Moscow 987 in Russian.