Some problems of analytic number theory

Ramachandra, K.

doi:10.4064/aa-31-4-313-324

Cited by 67 publications

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“…The main term is easily seen to bê Lc-2y^-2y^-i(a(^+ n-aoo), which is asymptotic to h 2 Vprovided that Y ^ x (7/12) +£ (these results are due to A. E. INGHAM and M. N. HUXLEY, see [3]). Thus we have following result.…”

Section: S=t^y^^+h-y)-^x-y))^(y+h)-^(y)) =T^y^+h-y)-nx-y))^(y+h)-^mentioning

confidence: 90%

“…The 0-term is easily proved to be 0 (h 1 Y exp (-(logx) 1/6)) provided h ^ y^^+s ^hese results are due to A. SELBERG and M. N. HUXLEY, see [3]). The main term is easily seen to bê Lc-2y^-2y^-i(a(^+ n-aoo), which is asymptotic to h 2 Vprovided that Y ^ x (7/12) +£ (these results are due to A. E. INGHAM and M. N. HUXLEY, see [3]).…”

Section: S=t^y^^+h-y)-^x-y))^(y+h)-^(y)) =T^y^+h-y)-nx-y))^(y+h)-^mentioning

confidence: 92%

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Two remarks in prime number theory

Ramachandra¹

1977

Bul. Soc. Math. France

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Section: S=t^y^^+h-y)-^x-y))^(y+h)-^(y)) =T^y^+h-y)-nx-y))^(y+h)-^mentioning

confidence: 90%

Section: S=t^y^^+h-y)-^x-y))^(y+h)-^(y)) =T^y^+h-y)-nx-y))^(y+h)-^mentioning

confidence: 92%

Two remarks in prime number theory

Ramachandra¹

1977

Bul. Soc. Math. France

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“…If we take as inputs the zero-density results in [12], (1.1), and [17], Theorem 3.2, and the zero-free region in [13], Theorem 2, then the procedure in [19] yields…”

Section: -75) It Follows Thatmentioning

confidence: 99%

The Parity Problem for Reducible Cubic Forms

Helfgott¹

2006

J. London Math. Soc.

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“…Unconditionally, using results towards the density hypothesis, it was previously known that there are cancellation of µ(n) in almost all intervals of length x 1/6+ε (a result due to Ramachandra [34]). One naturally wonders if it is possible to establish Theorem 1 in all intervals of length h ≍ √ X.…”

Section: Introductionmentioning

confidence: 99%

Multiplicative functions in short intervals

2016

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Abstract. We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the Möbius function we show that there are cancellations in the sum of µ(n) in almost all intervals of the form [x, x + ψ(x)] with ψ(x) → ∞ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of x ǫ -smooth numbers in intervals of the form [x, x + c(ε) √ x], recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of λ(n)λ(n + 1), with λ(n) Liouville's function, is non-trivially bounded in absolute value by 1−δ for some δ > 0. This settles an old folklore conjecture and constitutes progress towards Chowla's conjecture. Fourth, we show that a (general) real-valued multiplicative function f has a positive proportion of sign changes if and only if f is negative on at least one integer and non-zero on a positive proportion of the integers. This improves on many previous works, and is new already in the case of the Möbius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.

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Some problems of analytic number theory

Cited by 67 publications

References 0 publications

Two remarks in prime number theory

Two remarks in prime number theory

The Parity Problem for Reducible Cubic Forms

Multiplicative functions in short intervals

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