�ber eine Verallgemeinerung des Tschebyschef-Problems

Bentz, H.; Pintz, János

doi:10.1007/bf01215079

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A (X)= ~ (I) -~3 (X)1

Chebyshev Asserted In 1853 That Lira ~ (-1)(r-')/2e-pl~= -~1

We Formulate the Following Hypothesis For Dirichlet's Lfunct1

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1980

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“…Under some hypothesis (weaker than the Riemannr--Piltz hypothesis and fulfilled for every character modq with q < 25) we have shown in another paper [2] that the following statement holds for 1 0~<a<g:…”

Section: A (X)= ~ (I) -~3 (X)mentioning

confidence: 84%

“…L (s, •t) ~ O, Re (s) > 89 (2) with nonprincipal X1 mod 4. This was shown by HARDY, LITTLEWOOD and LANDAU in 1917/18 [6], [9].…”

Section: Chebyshev Asserted In 1853 That Lira ~ (-1)(r-')/2e-pl~= -~mentioning

confidence: 98%

“…x--+~ p 2 We always mention both hypothesis R 2 and It2 together, although It2 (as the weaker one) would suffice to prove the theorems, but it is easier to imagine the 'region' which is denoted by R2, and, in addition, some hypothesis of type R2 are used in related contexts. s(p,q,l,,12)logpexp(-(log2p)…”

Section: We Formulate the Following Hypothesis For Dirichlet's Lfunctmentioning

confidence: 99%

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Quadratic residues and the distribution of prime numbers

Bentz

Pintz

1980

Monatshefte f�r Mathematik

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Abstract. D. SHANKS [11 ] has given a heuristical argument for the fact that there are "more" primes in the non-quadratic residue classes mod q than in the quadratic ones. In this paper we confirm SHANKS' conjecture in all cases q < 25 in the following sense. If l, is a quadratic residue, 12 a non-residue modq, e (n, q, l,, ~z) takes the values -F 1 or -l according to n = I I or l~ mod q, thenfor 0 ~ ~ < 1/2. In the general ease the same holds, if all zeros ~ = fl + i y of all L (s, :~ mod q), q fix, satisfy the inequality fl" -7 "~ < l/4.

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