Estimates for Multiplicative Functions on the Set of Shifted Primes

Let fi, i = 1, ... k, be complex-valued multiplicative functions satisfying the conditions where i C, (*) and , (i = 1, ..., k), with some 0 < 1. Under these conditions we prove that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaeqiWdaNaaiikaiaadIhacaGGPaaaamaaqafabaGaamyq% aiaacIcacaWGWbGaey4kaSIaaGymaiaacMcacqWIQjspdaWcaaqaai% aabYgacaqGVbGaae4zaiaabccacaqGSbGaae4BaiaabEgacaqGGaGa% amiEaaqaaiaadIhaaaaaleaacaWGWbWefv3ySLgznfgDOjdaryqr1n% gBPrginfgDObcv39gaiuaacqWFMjIHcaWG4baabeqdcqGHris5aOWa% aabuaeaacaWGbbGaaiikaiaad6gacaGGPaGaey4kaSYaaSaaaeaaca% qGOaGaaeiBaiaab+gacaqGNbGaaeiiaiaabYgacaqGVbGaae4zaiaa% bccacaqGXaGaaeimaiaadIhacaGGPaWaaWbaaSqabeaadaWcaaqaai% aadogaaeaacaaIYaaaaiabgUcaRiaaigdaaaaakeaacaqGOaGaaeiB% aiaab+gacaqGNbGaaeiiaiaadIhacaGGPaWaaWbaaSqabeaadaWcaa% qaamrr1ngBPrwtHrhAXaqehuuDJXwAKbstHrhAG8KBLbacgaGae4x8% depabaGaaGOmaaaaaaaaaaqaaiaad6gacqWFMjIHcaWG4bGae8ha3J% habeqdcqGHris5aOGaai4oaaaa!863E!\\frac{1}{{\pi (x)}}\sum\limits_{p \leqq x} {A(p + 1) \ll \frac{{{\text{log log }}x}}{x}} \sum\limits_{n \leqq x} {A(n) + \frac{{{\text{(log log 10}}x)^{\frac{c}{2} + 1} }}{{{\text{(log }}x)^{\frac{\varrho }{2}} }}} ;\] moreover, if each fi satisfies (*) with C = 0, then there is 1 > 0, such that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaeqiWdaNaaiikaiaadIhacaGGPaaaamaaqafabaGaamyq% aiaacIcacaWGWbGaey4kaSIaaGymaiaacMcacqWIQjspdaWcaaqaai% aabYgacaqGVbGaae4zaiaabccacaWG2baabaGaamiEaaaaaSqaaiaa% dchatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab-z% MigkaadIhaaeqaniabggHiLdGcdaaeqbqaaiaadgeacaGGOaGaamOB% aiaacMcacqGHRaWkdaWcaaqaaiaaigdaaeaacaWG2bWaaWbaaSqabe% aatuuDJXwAK1uy0HwmaeXbfv3ySLgzG0uy0Hgip5wzaGGbaiab+f-a% XlaaigdaaaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaaiikaiaabY% gacaqGVbGaae4zaiaabccacaWG4bGaaiykamaaCaaaleqabaGae4x8% deVaaGymaaaaaaaabaGaamOBaiab-zMigkaadIhacqWFaCpEaeqani% abggHiLdaaaa!7A93!\\frac{1}{{\pi (x)}}\sum\limits_{p \leqq x} {A(p + 1) \ll \frac{{{\text{log }}v}}{x}} \sum\limits_{n \leqq x} {A(n) + \frac{1}{{v^{\varrho 1} }} + \frac{1}{{({\text{log }}x)^{\varrho 1} }}} \] holds, where 3 < v < logAx. As a corollary we prove some results about the mean-value of multiplicative functions.